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Over Constrained Linear Systems

Posted by bilsef 
Over Constrained Linear Systems
May 03, 2012 12:08AM
Are you having trouble getting your carriages to ride smoothly on their axes? The root cause of the problem is most likely that the axis is over constrained. This seems to have gotten worse with the popularity of linear bearings and the propensity to put them on all four corners of a carriage. It's probably because cars have four wheels that we are accustomed to seeing it that way. I will attempt to explain why that is wrong and what can be done to fix it.

First of all, you should understand that it's virtually impossible to align two shafts perfectly parallel to each other. Even if you are able to get them close, they won't stay that way. Change in temperature, motion, and wear will cause them to move out of alignment.

When you put two bearings or bushings on a single shaft that is fixed on both ends (a vector), you have constrained 4 of the 6 possible degrees of freedom for that axis. There are different names for these degrees of freedom, but I like to call them Longitude (X), Latitude (Y), Elevation (Z), Pitch (rotation around X), Roll (rotation around Y), and Yaw (rotation around Z) ( [en.wikipedia.org]) ). So if we are talking about the X axis, you have constrained everything except Pitch and Longitude. You want the carriage to move freely in the X direction, so the only remaining constraint to add is for Pitch. To do this, we use a second shaft that is (mostly) parallel to the first shaft. If we only contact this second shaft on the top and bottom edges, we can stop rotation about the X axis. As long as you don't touch any other edges of that shaft, the carriage will move freely. If the second shaft is out of alignment in the Y direction, it will not have any affect on the motion. If it is out of alignment in the Z direction, the carriage will pitch as it moves left and right. To reduce this pitching, you only need to align the two shafts in the Z direction, which is a much simpler task.

With the normal vector (master shaft) and the single point (idle shaft contact), we have created a plane (the carriage surface) ( [en.wikipedia.org]) ). If we try add one more contact point on the idle shaft it will have to conform to that plane. Chances are it will not and there will be binding or something will have to bend. So resist the temptation to add a second idler.

Ideally the idler contact is centered between the two bearings to balance the load, but if the load is light, it is not that important. In the typical Mendel design a centered idler will get in the way of the hot end. So just put it off to the side where the weight of the extruder motor is. For the Y axis it can usually be centered without any problem.

The belts driving the axis should be attached closer to the side of the carriage that has the two bearings (the master). This way there will be less sideways load on the bearings due to the moment of inertia ( [en.wikipedia.org] ). The further away you get from the shaft, the longer the moment arm is.

If you have a X carriage with bolt on bearing holders, like one of the Supa-Flat varieties ( [www.thingiverse.com] ), you can try this idler attachment to see how much smoother your X travel becomes ( [www.thingiverse.com] ). Here is a short video demonstration of a similar change to a MendelMax Y carriage ( [plus.google.com] ).

The Z axis in the typical Mendel design is really two independent systems that move in sync with each other. The best way to eliminate binding is to not have them tightly coupled together. These X ends from kludgineer address the problem: [www.thingiverse.com]
Re: Over Constrained Linear Systems
May 03, 2012 04:01AM
Nice summary of the subject. Perhaps it would be also useful in the wiki? I'm thinking it might be found there more readily than here in the forums, but I don't know if people search the wiki nor where it would be best placed. Is there a section for "Engineering concepts" or "Things worth considering"?


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garyhodgson.com/reprap | reprap.development-tracker.info | thingtracker.net
Re: Over Constrained Linear Systems
May 03, 2012 11:19AM
"It's probably because cars have four wheels that we are accustomed to seeing it that way."

In an ideal world it is the better way to design it. It looks better on drawings, my favorite CAD program shows it lined up better, and my favorite motion analysis program shows it to work the best.

Of course we don't live in an ideal world and simulations don't always predict the real world or practical experience. I see new engineers over constrain things in designs all the time.

Anyway,
Great write up, and it sums up some 50 pages of bearing placement from the machine design book quite nicely and accurately!
Re: Over Constrained Linear Systems
May 19, 2012 12:52PM
Your description of degrees of freedom is a sound one. It was one that was incorporated in the original reprap design. See below.

Reprap axis design video

This is a sound theory, but in the real world was not found to be the whole story as suggested by Chris above. The original designers thought that the mechanism would not be accurate enough for parallel rails, but this has been found to be untrue. Some clever fellow tried LM8UU bearings and found them to work very well. This is because the natural flexibilty of the parts and tolerancing of the bearings gives enough lateral movement for it to work. This is what is omitted in any theoretical DOF study and is understandable particularly when the designers came from an academic background.

In the machine tool industry parallel rails are used most often as the accuarcy of machined parts enables a sufficient degree of parallelism for the mechanisms to work without problems. You also get an additional benefit of greater load capability and higher accuracy provided by the additional degrees of constraint.
Re: Over Constrained Linear Systems
May 19, 2012 01:53PM
The funny thing about the video is that the Mendel X and Y axes are over constrained. The side with 180 bearings should only have one pair in the middle. Z needs two because there is nothing rigid to stop the X axis twisting slightly along its length.


[www.hydraraptor.blogspot.com]
Re: Over Constrained Linear Systems
May 21, 2012 03:56PM
martinprice2004 , thanks for the link to the video. I knew the original design was not so over constrained, but hadn't seen that explanation.
What you are describing as "natural flexibilty of the parts and tolerancing of the bearings" is just slop that has to be designed into the system in order to allow it to work.

After using my MM 1.5 for a while, I decided to re-design the X ends to use a 180 degree bearing for the Z axis also. This one clamps the X rods on both ends (like other Mendel designs). I also moved the X belt below the X carriage, but this is just a personal preference. I will test this in the next few days.

[www.flickr.com]

nophead, I was thinking about filing flats on one end of both X bars to prevent twisting, but maybe it's better to just stick with two 180 degree bearings for Z.
Re: Over Constrained Linear Systems
May 22, 2012 12:49AM
Putting flats on the X bars will stop them rotating but that isn't an issue. The clamps stops them rotating, but you can easily twist one end of the axis relative to the other because over that length the bars can flex a little. That means you need 360 bearings at each end to properly constrain it.


[www.hydraraptor.blogspot.com]
Re: Over Constrained Linear Systems
May 22, 2012 07:56AM
On my reprap variant I have used tie wraps to hold the x axis rails in place as well as the linear Z axis bearings. It was an experiment to reduce the number of fasteners and simplify the build. The bars pass through reamed holes and the Z axis bearings sit in a simple V groove. The tie wraps make an excellent mechanical joint and is comparable to a bolted one if designed correctly.

I can't compare it to a conventional reprap for part quality as I don't own one, but I must say the parts it produces are accurate and square. I would be confident to reproduce another reprap on it without problems.

Perhaps there is some way you could design one of the rails and use a cable tie to give you a little compliance in the direction you require.
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