The math behind Z ribbing explained

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The math behind Z ribbing explained

Postby RetireeJay » 2013-Aug-Thu-10-Aug

EDIT 7 Jan 2014: Yikes, this whole thing is hot air. Truth is, you won't see ribbing caused by non-integer layers in Z. viewtopic.php?f=21&t=5983 However, the math is interesting, and if you want to understand better how to calculate and manipulate calibration factors, read on... :D

There have been many posts about ribbing, and there are online articles about it too. But in this post I want to try to explain it in a way that most people with the skills to do high-school algebra will understand the mathematical roots of the discussion and be able to re-create their own calculations if they want. Understanding is always better than blind copying, no? ;)

Basically there are three factors that are inter-related and you can only choose two out of the three to be exactly what you want arbitrarily.

1) Calibration: steps per millimeter. This affects the accuracy of the height of your objects. Most people seem to insist on the "ideal" 2267.72 number of steps, although in fact the true number would go to an infinite number of non-repeating digits. Here's what my calculator produces: 2267.7165354330708661417322834646. (In fact, I use 2260 steps per mm, an error of 0.324%, for reasons that will be clear in a moment.)

2) Layer Height: Ideally we'd like to think that we can specify any arbitrary layer height, but the fact is that the stepper motors move in certain increments of distance. We can specify any layer height we want in Slic3r, but only certain choices will give us no ribbing. Also, it would be nice if we could specify the layer height with an exact number, not a rounded-off number that is an approximation.

3) Absence of ribbing: The printrbot calculates in floating-point arithmetic how high the printhead should be for each layer. Then it translates that into the number of steps required to reach that height. If the theoretical height for a single layer is not exactly equal to a certain number of steps, then the printrbot must round off the number to get an integral number of steps. However, the errors build up layer by layer until you reach a layer where the rounding routine either adds a step or subtracts a step in order to make the actual height of the printhead as close as possible to the theoretical height at that layer. This is the cause of ribbing: every so many layers (often in the range of 6 to 10 layers) you will see a layer that has one extra step or is missing one step. It's not a huge difference, but in many models it can be noticeable.

So what to do?

When you have 5/16 rod and use the 2267.72 calibration, then in order to have layer heights which can be represented with perfect accuracy in a limited number of decimal digits AND result in layers which use an integral number of motor steps, then you are constrained to layer heights which are multiples of 0.031750 mm. This series includes layer heights like 0.190500, 0.222250, 0.254000, 0.285750, 0.317500, 0.349250, 0.381000, etc.

[On the other hand, if you are willing to accept the 0.34% error in calibration (who's really going to notice?) then you can use layer heights of 0.2, 0.25, 0.3, 0.35, 0.4...]

[When you change to a metric rod, things get a lot easier, but I'll leave that for the reader. The principles here will apply.]

In reality, all the math behind this is fairly straightforward.

DATA:
The 5/16 threaded rod has 18 turns per inch.
One inch is exactly 25.4mm
The motors have 200 steps per turn, but with 16X microstepping, it is 3200 steps per turn.

CALCULATIONS
Thus to move one inch, the motor turns 18 * 3200 steps. To move one mm it turns (18 * 3200) / 25.4 steps. That's where the 2267.72 comes from.

Now here's how we maneuver it to get the 0.03175:

We want a layer height which is BOTH an integral number of steps AND which also can be expressed exactly in a small number of decimal digits.
So [Layer Height(mm) / Distance per Step(mm)] should be an integer.

Now Distance per Step is the inverse of the figure we found out earlier: it's 25.4 / (18 * 3200) mm per step.

Let's regroup numbers in the denominator:
Distance per Step = 25.4 / (18 * 3200)
= 25.4 / (18 * 4 * 800)
= 25.4 / (72 * 800)
= (25.4 / 800) / 72
= 0.03175 / 72.

So layer heights that are multiples of 72 steps will come in integral multiples of 0.03175mm.

For example, a layer height of 0.254mm = 576 microsteps = 8 * 72 microsteps.

Now if on the other hand we said arbitrarily that there were 2260 steps per "Jay's millimeter," then our distance per step would be
1/2260
= 1/(113 * 20)
= 0.05 / 113

So layer heights that are multiples of 113 steps will come in integral multiples of 0.05 Jay's millimeter. Convenient, no? I can specify a layer height of 0.30 Jay's millimeters and use exactly 678 steps. (In fact the actual layer height using the 5/16 rod will be 0.2989792 mm - pretty darn close to 0.300mm)

Congratulations if you've made it this far! Now you're an expert on the calculations behind z-ribbing and how to avoid it.
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The math behind Z ribbing explained

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Re: The math behind Z ribbing explained

Postby KDog » 2013-Aug-Thu-12-Aug

Thanks for this well thought out and precise explanation Jay. I got my PBJ a few months ago and noticed when calibrating that the default value for the z-steps was 2276 rather than 2267. Wondering if anybody else noticed this. Do you think there was a reason for this value or did somebody switch the digits?

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Re: The math behind Z ribbing explained

Postby RetireeJay » 2013-Aug-Thu-12-Aug

It might be a typo, or it might be a value that somebody came up with empirically.

I can pretty much guarantee that if you try to calibrate empirically (that is, by measuring) you won't exactly match the calculated values. But if you don't have a fully-equipped measurement laboratory, you probably should stick with the calculated values based on the geometry of your machine's actuators to get started. Later on, if measurements of objects you print are not what you want them to be, then tweak the calibration so your printed objects match your specification.
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Re: The math behind Z ribbing explained

Postby iesvilla » 2013-Aug-Fri-11-Aug

Thanks... like A LOT. This is helping me to increase the quality of my prints by a noticeable margin. Great work! :D
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Re: The math behind Z ribbing explained

Postby mdfast1 » 2013-Aug-Fri-16-Aug

I say we sticky it. Z ribbing is the bane of a lot of newbies.
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Re: The math behind Z ribbing explained

Postby andersonmatt1125 » 2013-Sep-Tue-04-Sep

I just want to make sure I understand this. You're saying that if I calibrate my Z steps to 2260, I will completely eliminate Z-ribbing, but at the cost of a .3% calibration error?

This seems like a good deal to me, but it opens up another question. You mention that people suggest an "ideal" Z steps to be 2267.72. If that's so, then why does the printrbot getting started guide 2 (I think it's the older, but more detailed one linked to at the bottom of the current one) suggest to use 2387.0719? This doesn't seem to match up with anything else.
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Re: The math behind Z ribbing explained

Postby RetireeJay » 2013-Sep-Tue-09-Sep

andersonmatt1125 wrote:I just want to make sure I understand this. You're saying that if I calibrate my Z steps to 2260, I will completely eliminate Z-ribbing, but at the cost of a .3% calibration error?


Not exactly. What I'm saying is that if you use 2260, you can choose layer heights that are multiples of 0.05mm and get no z ribbing. Every z calibration value has certain layer heights that will yield no ribbing; 2260 gives you non-ribbing values that are easy to remember and can be written with a small number of digits and no rounding.

This seems like a good deal to me, but it opens up another question. You mention that people suggest an "ideal" Z steps to be 2267.72. If that's so, then why does the printrbot getting started guide 2 (I think it's the older, but more detailed one linked to at the bottom of the current one) suggest to use 2387.0719? This doesn't seem to match up with anything else.


Yes, the original getting started guide had some queer numbers. I think that they were in a rush to get the guide published, and the person doing the guide took empirical measurements. It's hard to take empirical measurements of motion like this with an accuracy better than 1%, although the 5% error is the guide seems a little extreme. But if you look at the first part of the post, it's very clear that the geometry of a 5/16" - 18 rod and a motor with 3200 microsteps gives you 2267.717..... steps per millimeter. You would only want to change that if you had reason to believe that your rod was systematically imperfect.
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Re: The math behind Z ribbing explained

Postby andersonmatt1125 » 2013-Sep-Tue-11-Sep

Thanks so much. I'm going to change the z-calibration to 2260 and see if it helps my prints!
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Re: The math behind Z ribbing explained

Postby plexus » 2013-Sep-Tue-12-Sep

Can someone post the exact same print that shows the effect of shifting the Z cal setting for integral layer heights? i'd like to see how this actually affects print quality.
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The math behind Z ribbing explained

Postby Nickies08 » 2013-Sep-Tue-12-Sep

ImageUploadedByTapatalk1378230683.821418.jpg
ImageUploadedByTapatalk1378230699.105774.jpg


Plexus; in the thread I have "the taller, the wider" you can see some prints I did in white ABS with z-banding and then some I did after in Black ABS after making the changes Jay recommended. They are the same model just different layer settings. I attached two pictures so I hope they work because tapatalk has been troublesome on that feature.
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Re: The math behind Z ribbing explained

Postby plexus » 2013-Sep-Tue-20-Sep

I'm using ACME lead screws. they are 1/4"-16. my measured Z cal is 2028.95. so I worked out 2020. that's from 1/20 resolution (0.05mm) into an even number to get as close to 2028.95 as possible. that's 2020 = 20 * 101. did I understand the idea correctly?

I printed out a cylinder with 2028.95 and then with 2020. no difference in the quality of the print. I examined using a loupe. neither had any visible amount of Z ribbing. not sure what that means.

Also, looking at the pictures Nickies08 posted, the ribbing on the top one (white plastic) does not look like calculation ribbing - each bump gradually gets more pronounced and then back again. if it were ribbing due to calculation error wouldn't the ribbing be more abrupt? that looks like text book uncentred Z threaded rod ribbing to me (plus some X axis slop).
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Re: The math behind Z ribbing explained

Postby RetireeJay » 2013-Sep-Wed-11-Sep

I agree with the use of 2020, and your calculation.

Let's look at the theoretical:
Steps per mm = 16 * 3200 / 25.4 = 2015.7480
Distance per Step = 25.4 / (16 * 3200)
= 25.4 / (16 * 4 * 800)
= 25.4 / (64 * 800)
= (25.4 / 800) / 64
= 0.03175 / 64.

So with the theoretically perfect calibration, you could still use layer height multiples of 0.03175, getting 64 steps per layer.

If you use 2020 steps/mm, then as you say you have
Distance per step = 1/2020
= 1/(20 * 101)
= .05 / 101

So your .05mm layers will have 101 steps per layer, exactly as you said.

I'll admit that perhaps the pictures posted by Nickies08 may represent an extreme case, or may even show the result of more than one change to the setup between samples... When I tried to look at the influence of Z ribbing it was much more subtle but still definitely visible. However I'm not sure I still have those samples around...

You can set up a spreadsheet where you have a column for layer number (1...100), a column with your requested part height (your requested layer height * layer number), then another column with the number of steps the Printrbot will use (Round(requested part height * calibrated steps/mm),0), and then another column where you subtract the steps for the current layer from the steps for the previous layer. Ideally, this steps per layer column always has a constant number - but typically it does not. So you can look for the pattern: does the number of steps bounce every 6 layers, or every 10 layers, or what? Then you can look for that pattern in your printed part.
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Re: The math behind Z ribbing explained

Postby plexus » 2013-Sep-Wed-13-Sep

Thanks Jay. but I don't see any difference in my prints as I mentioned so I am just going to go back to the calibrated values. I basically get no ribbing either way. YMMV
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Re: The math behind Z ribbing explained

Postby RetireeJay » 2013-Sep-Thu-13-Sep

Plexus, you didn't say what layer height you were using. It is the convergence of two factors (calibration and layer height) that give rise to ribbing, and the ribbing will take different patterns depending on the combination.

If you were using 0.4mm then your use of 2028.95 for z cal would give you "ribbing" every other layer - which would be hard to notice. And of course no ribbing with the cal of 2020. But if you printed samples at a layer height of 0.41mm, then I would expect you to see visible ribbing with either the 2020.95 or the 2020 calibration. The first will give you a thin layer about every 8th layer and the second will give you a fat layer about every 5th layer.
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Re: The math behind Z ribbing explained

Postby plexus » 2013-Sep-Fri-00-Sep

Oh. I printed at 0.3mm
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Re: The math behind Z ribbing explained

Postby RetireeJay » 2013-Sep-Fri-08-Sep

In that case, your ribbing at calibration 2028.95 was one out of 3 - still a little hard for normal vision to see the pattern. If you tried 3.05mm layer thickness with either of your calibration choices, you would be more likely to see the ribbing. It will be at an interval of 6 layers (2028.95) or 10 layers (2020).
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Cable chain to reduce probability of fatigue failure in wires
E3D V5 Hot End, 0.4mm nozzle, also 0.8 and 0.25 in use occasionally
PB fan mount + 40mm fan -- using printed mount adapter, not the E3D supplied fan
Injection molded extruder gears
Optical Z "endstop" (custom designed and built)
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Re: The math behind Z ribbing explained

Postby plexus » 2013-Sep-Fri-09-Sep

I understand. it seems counter-productive to go searching for this ribbing when normally my prints don't have it with either setting. I think I will leave it as-is but this is good info if there is a ribbing problem and it can't be traced to dynamic hardware.
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Re: The math behind Z ribbing explained

Postby hugerobots » 2013-Sep-Sat-12-Sep

Any possibility that the rate at which the goop* is extruded will inevitably give you ribbing?

I don't see any chance of 0 ribbing because it's coming out of the hot end, ultimately as a globulous cylinder that will always have rounded side edges. 3mm to molecular, rounded edges on each layer are just basically going to happen. Unless there's some calculation involved to negate the propensity to be pushed out like a liquid in gravity.


* scientific latin terminolillloquy
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Re: The math behind Z ribbing explained

Postby RetireeJay » 2013-Sep-Sat-16-Sep

hugerobots wrote:Any possibility that the rate at which the goop* is extruded will inevitably give you ribbing?

I don't see any chance of 0 ribbing because it's coming out of the hot end, ultimately as a globulous cylinder that will always have rounded side edges. 3mm to molecular, rounded edges on each layer are just basically going to happen. Unless there's some calculation involved to negate the propensity to be pushed out like a liquid in gravity.


* scientific latin terminolillloquy


Every print made with Printrbots has layers, yes. The topic under discussion is a phenomenon where the layers are not equal. It's as if you are looking at a brick wall and every 6th course of bricks is offset a tiny bit - out of the plane of the other bricks.
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PB fan mount + 40mm fan -- using printed mount adapter, not the E3D supplied fan
Injection molded extruder gears
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Re: The math behind Z ribbing explained

Postby rho-wan » 2013-Sep-Sat-16-Sep

So basically if you use the RepRap calculator with 400 steps per revolution you are fine anyway, without having to calculate the values :)
But thanks for explaining exactly!
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Re: The math behind Z ribbing explained

Postby RetireeJay » 2013-Sep-Sat-18-Sep

rho-wan wrote:So basically if you use the RepRap calculator with 400 steps per revolution you are fine anyway, without having to calculate the values :)
But thanks for explaining exactly!


400 steps per revolution? I don't understand. Our motors are set up with 16X microstepping, so we have 3200 steps per revolution. But the steps per revoluion is a constant; it's the Z calibration and layer heights that are the subject of this post.

The interaction of Z calibration and layer thickness is inescapable. You simply cannot freely choose any layer height you want (even with metric rod) unless you are willing to live with the possibility of z ribbing. And the correct layer heights are tied to the Z calibration.

That said, some calibration values allow you convenient layer heights, like multiples of 0.05mm. That's why I use 2260 steps per mm.
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E3D V5 Hot End, 0.4mm nozzle, also 0.8 and 0.25 in use occasionally
PB fan mount + 40mm fan -- using printed mount adapter, not the E3D supplied fan
Injection molded extruder gears
Optical Z "endstop" (custom designed and built)
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Print on glass with Scotch Craft Stick or other glue stick
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Re: The math behind Z ribbing explained

Postby rho-wan » 2013-Sep-Sun-14-Sep

You are right, the numbers don't match, but I noticed that if you go to that layer height calculator and you input 400 steps per revolution (the maximum allowed by the calculator), you should just check which layer thicknesses are multiples of 9 steps.
Well, at the end it's easier to use your formula.
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Re: The math behind Z ribbing explained

Postby PxT » 2013-Sep-Thu-13-Sep

andersonmatt1125 wrote:This seems like a good deal to me, but it opens up another question. You mention that people suggest an "ideal" Z steps to be 2267.72. If that's so, then why does the printrbot getting started guide 2 (I think it's the older, but more detailed one linked to at the bottom of the current one) suggest to use 2387.0719? This doesn't seem to match up with anything else.


I wrote the original guide. At the time (almost 18 months ago - an eternity!) I didn't understand all this stuff very well and I published a number that worked acceptably for me at the time. I don't recall exactly where it came from. It might have been based on a measurement taken from my machine or it may have come from somewhere on the web. I was in a bit of a rush to get any kind of documentation up since there was really nothing available at the time.

In any case I don't think the version with the 2387 value is still published anywhere, there were subsequent revisions of the document which corrected it to a more appropriate value.
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