x-, y-bars too weak?

I calculated the expected bending of the x and y bars
with this model (free endings, force in the middle, both ends supported)

      F
      v
=============
^           ^
/---- l ----/

with this formula i found in my brothers static book:
b = (F * l^3)/(48*E*J)
J = 0.7854*r^4

b : maximum expected bending in mm
F : force in kN
l : bar length in mm
E : elastic modulus in kN/mm^2 (steel = 350, aluminium=70)
J : moment of inertia
r : Radius of the bar in mm

For a mendel with a heated bed I used:
F = 0.01 (2kg y-carriage /2 bars), l=400, E=350, r=4 and got
b = 0.31mm
So the bars should bend almost one filament diameter.
with r=5mm => b=0.12mm
with r=6mm => b=0.06mm

To get the bending below 0.1mm the y-bars should be 12mm in diameter.
I know bending will be less in the real world (force not punctual, force at the and not zero, y-carriage ligther). But on the other hand, it can't really hurt to make them stronger, and it looks like they are on the weak side.

My darwin has longer bars, and much, much heavier axes, and there is no noticeable deflection. With the 8mm steel bars I'm using, 1m clamped at one end is very difficult to bend even by a fraction of a mm. I'm not sure why your calculations give such bendiness, but I don't think its much of a problem in practice. I think a better approach than stronger bars would be lighter axes if people are having this problem, 12mm steel bar is difficult to get, expensive and heavy.

Standard gravity is taken as ~9.8 N/kg, so 2kg would be ~19 Newtons. or .02 kN? Hmm. I was just sure that was where the error was.

There is an error somewhere - gravity is the least of the forces the frame is subject to. The stepper motors slamming the head from one side to the other is MUCH stronger, and doesn't deflect significantly.

---

--
I'm building it with Baling Wire

jgilmore Wrote:
-------------------------------------------------------
> Standard gravity is taken as ~9.8 N/kg, so 2kg
> would be ~19 Newtons. or .02 kN? Hmm. I was just
> sure that was where the error was.
>
> There is an error somewhere - gravity is the least
> of the forces the frame is subject to. The stepper
> motors slamming the head from one side to the
> other is MUCH stronger, and doesn't deflect
> significantly.

Are you positive you are not correct? .1mm deflection would probably not be noticed in a printed object, and would probably be irrelavent given that the filament itself has greater deviation from moment to moment as the pressure inside the melt chamber and motion on the axes change. I also don't think the deflection is linear, I.E, to deflect it 0.2 mm may require much greater than twice the force.

For context, .1mm is about the thickness of a paint of coat, length of a dust particle, or the thickness of office paper... (In context with the filament itself, which should be a little less than the thickness of a credit card.)

Funny that this came up; I was thinking of running these numbers myself the other day.

I have been through the OP's calculation and everything looks good for his assumptions, et cetera based on my experience as an engineer. He's using the correct deflection formula, calculating moment of inertia properly, et cetera. Even his unit conversions and consistency look good.

I will say that when I run the actual calculation myself, I get a different answer - 0.19mm. OP may want to double-check his math, or perhaps I need to.

Regardless of the answer, one possibility for any disparity between model and actual is the fact that he modeled this as a point load. In reality, the carriage is not a point load. Not only that, but the loads themselves are connected by the carriage chassis. This can make a pretty significant difference.

Edited 1 time(s). Last edit at 05/05/2010 03:19PM by plasmator.

The calculations are roughly correct give or take a few microns (I'm a mech. engineer), but the simple bending equation doesn't take the real world into account and can have an error of +/- 30% which in this case is about +/- 0.1 mm.
And your elastic modulus of the steel is a bit high, it should be around 200 GPa.

So in theory you are looking at a bar deflection of about 0.5 mm as the worst case. In reality, at least for mine, my worst bar deflection of 8mm diameter 304 stainless is 0.23 mm according to a laser measurement device and visually noticed as about a hair's width.
For me, it is a tolerable deflection.

You also have to take the whole system into account. So the y-axis beam may deflect a half a millimeter or so, the z axis height might be off by 1 or 2 millimeters. Sum up all of the errors caused by bending, loose joints, motion, jerking, banging, etc. and you will probably have an error of +/- 1cm or more,
yet in the real world the nozzle stays at a pretty constant height over the bed.
The height differences on my machine from nozzle to bed are +/- 0.004 mm.

A larger diameter bar would elimate some of the bending, but ultimately you do have to trade off the ideal design for what is available at a reasonable cost.

jgilmore Wrote:
-------------------------------------------------------
> Standard gravity is taken as ~9.8 N/kg, so 2kg
> would be ~19 Newtons. or .02 kN? Hmm. I was just
> sure that was where the error was.
There are 2 bars

> There is an error somewhere - gravity is the least
> of the forces the frame is subject to. The stepper
> motors slamming the head from one side to the
> other is MUCH stronger, and doesn't deflect
> significantly.
These forces are in the dircetion of the bars, that makes a huge difference.

plasmator Wrote:
-------------------------------------------------------
> I will say that when I run the actual calculation
> myself, I get a different answer - 0.19mm. OP may
> want to double-check his math, or perhaps I need
> to.
I wrote the wrong elastic modulus in the post, which criswillson spotted correctly. But in the calculations I used the right one

At least now I know why taking aluminium bars for my machine was a bad idea. After reading your posts I think it's not worth the effort for extruding, as one would have to redesign quite a bit of the machine. But if you want to mill with a mendel (with much higher z-axes forces) it's probaply worth it.

Milling and printing are definitely to different things when it comes to machine designs. To mill with a mendel I would increase the bar diameters to at least 12 mm, but probably more like 25 mm (those are non calculated off the cuff guesses).

One important thing from this is that you learned how to do a theoretical beam deflection and you got the correct answer.

And if you want to go with Aluminum bars for printing, then you should probably start your calculations around a 30mm diameter bar for 0.1 mm of bending.

I think Mendel will be hopeless at milling, even with bigger bars because there is no diagonal bracing, so you can easily displace the tool in the direction of the x-axis.

---

[www.hydraraptor.blogspot.com]

Alright back to the drawing board then. My 5mm rods Ive been getting out of inkjet printers aren't going to be usable on x and y.

I was watching one of Adrian's talks and he seemed to think milling with a flex shaft dremel was possible for pcb production.

Ah rather then cover 300+ mm while doing circuit boards, how about some form of restricted work area?

Maybe a temporary brace while doing the milling of circuit boards or heck gearing to increase the acuracy over a smaller area?

I bought one of those Dremel flexible shafts only to find it has end play

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[www.hydraraptor.blogspot.com]

For Bertha the CNC machine I had to use 4 8mm bars for the X and Y axis due to bar flexing with four 8mm slider bars the problem vanished.



For the Big Bertha CNC 4'x 2' I have 20mm hardened steel slide bars

8mm bars are quite adequate for Darwin and Mendel neither machines are stiff enough to mill maybe light engraving like PCB engraving ~ I think you could just about get away with using a bowden drive with a dremel type drill mounted separately from the frame.

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