Excessive radial clearance in my linear motion axis, necessary because of the tolerances I could achieve using conventional woodworking techniques, was the major cause of inaccuracies and non-repeatability in my system. We learned recently in class that appropriate preload is a feasible approach to eliminate radial clearance and improve geometric repeatability without compromising overall bearing stiffness.
There is a large variety of possible ways to build preload into a sliding contact linear bearing. I decided to explore the use of an elastomeric compliant layer in the slider since it seemed like a concept I could eventually incorporate into my desk. Few suitable compliant materials have the wear and frictional properties to be good bearing materials, so this compliant layer will be sandwiched between the original wooden slider and a new bearing plate made of a wear-resistant material.
All of the repeatability and stiffness benefits of preloaded bearings only hold if the preload is never completely relieved under normal operation. I estimated that the primary source of aberration in the boxway is due to local imperfections in the side rails due to the poor surface finish achievable in the oriented strand board I used to build them. To the first order, I estimated that these imperfections have a characteristic dimension of approximately 0.1 mm. In order to ensure some preload is retained through the entire travel, I invoked St. Venant’s Principle and designed for a preload displacement of 10 mm.
Using these values, I built a simple spreadsheet that predicts the marginal amount of actuation force due to the preload and specifies the geometry of individual parts to help support fabrication. For a hypothetical material with a modulus of 0.001 GPa (based off of linoleum), the appropriate displacement would result in an additional 140 N of actuation force required. Since my toy linear motion axis does not drive anything other than its internal resistance, I figured my stepper motor would be able to move it without too much trouble.
I had to change out a flat tire on my bike this weekend, and I took the opportunity to take a closer look at the tire levers I was using. These tire levers from Park Tool has nifty feature where they snap together. This is very useful in keeping them together in a messy tube change kit.
The way they attach to each other is through a Lego-esque snap-fit feature molded into the part. Unlike Lego, however, the boss on the bottom lever only makes contact with the mating lever at the two cylindrical faces. And therein lies the reason why these levers don’t align perfectly every time, unlike Lego. As we saw a few weeks ago, Lego bricks make use of elastically averaged interfaces to achieve amazing performance (at least for an injection-molded plastic brick). Like Lego bricks, these levers are held together by local elastic deformation; but unlike Lego bricks, these only make contact at a small number of points (2 in this case). Obviously, averaging over a small number of contact points results in significantly lower repeatability.
Another issue with this interface is that it has a large amount of wiggle in it. That is, not only do the levers not line up the same way every time you snap them together, but they also rotate appreciably (~0.5 mm) relative to each other while engaged. This is a result of using only 2 contact points. As we know, 3 points of contact are necessary to fully constrain a planar part (the local elastic deformation provides the preload in this case). Unfortunately, this wiggle does detract slightly from the sense of quality you otherwise get from using these otherwise excellent tools.
Well, if anyone from Park Tool is reading this: when you next get a new mold, include a third point of contact!
After ruminating on the overall architecture of my desk for the past couple of weeks, I have decided to commit to a desk-mount design with a single linear motion axis. My reason for pursuing a desk-mounted design is so that the new device would fit into current and future spaces that I would live in. I anticipate living in pre-furnished spaces in the near future, so an adjustable desk that complements an existing desk is preferable to an additional piece of free-standing furniture.
I have also decided to limit myself to a single axis of motion in order to control costs and mechanical complexity. In my experience using desks, I have had little occasion to desire adjustment other than height. I am currently considering a single-column cantilever design as well as a twin-column design. Read more about these concepts below.
I am setting a total allowable error of 5 mm along the height dimension for my desk. 10% of this is allocated to the actuator (I think cheap bearings and structural members would contribute significantly more error than purchased leadscrews), and the remainder is evenly split between bearings and structure. As I mentioned previously, user experienced is degraded far more significantly by “free wiggle” geometric errors than load-induced deflections that are associated with significant resistance.
This is an early concept that I started off with, and it has stayed around through the process largely because of its simplicity. Additionally, there is something visually appealing (to me) about a cantilevered desk with a substantial support column made from wood.
The drawbacks are primarily associated with the relatively long cantilever length, which natural translates to larger deflections and higher forces at the bearing.
This alternative design has the tabletop simply supported (an idealization) between two bearings each running in a column. These columns are located centrally at each of the sides of the desk. The advantage of this design is that it halves the moment arm for the worst-case scenario where all of the design load is concentrated at one corner of the desk. This translates to lower forces felt at the bearing and smaller deflections in the tabletop.
However, the need to double up on structural material and drive mechanisms (need to somehow actuate both sides to prevent racking) would increase cost. Additionally, the visually interesting bearing columns are not moved to the sides and out of direct view for the user. And finally, the columns may turn into a nuisance by reducing elbow room.
In the interest of time, I am going to focus on the single-column design for my exploration this week. The twin-column design will serve as a backup that I can come back to if I run into unanticipated and terminal problems with the single-column design.
See my first-order error budget for the single-column design here.
Quick Note on Self-Locking Bearing Concept
Avid readers (there are not that many…) of this site will remember reading about an idea I had to exploit the binding characteristics of sliding bearings to achieve self-locking. The primary reason for doing that was to allow the use of a more compact and lighter belt or chain drive system. Unlike leadscrews, belt or chain drive systems generally don’t have sufficient friction in the drivechain to resist substantial forces when powered off.
I spent some time this week analyzing a concept that has a chain attached some distance away from the center-of-stiffness of the bearing. The idea was to design the bearing so that the weight of the table and things piled on top would cause the bearing to bind up and resist moving downwards. When the actuator is switched on, it pulls on the chain to relieve part of the binding moment, thereby releasing the table to move upwards. Unfortunately, my analysis showed that the reciprocal action to this — when the user wants to move the table downwards — actually increases the binding moment and causes even more friction. I did some what-ifs scenario in a spreadsheet and found that my actuator would have to exert thousands of Newtons to overcome the friction and move the table downwards. This is not only difficult to achieve using a small stepper motor, but also very very difficult to achieve without breaking the structure or bearings due to the large forces involved. Ultimately, I think there are too many drawbacks to this idea for it to be worth pursuing further, so back to leadscrews it is.
The single-column design is made up of 4 structural elements:
Since this is a desk, the sensitive direction is vertically up and down.
I am making this error budget assuming my entire design load (20kg) is concentrated at one corner of the tabletop. This is a worst-case scenario that is unlikely to happen in use, but gives conservative error estimates that would guarantee performance. I can back off on the worst-case scenario for further design refinement if it turns out to demand excessive material or cost.
The cantilevered tabletop is under bending both across the depth of the table and across half of the width. We can approximate their joint contribution using superposition, although this may not be exactly accurate due to shears in the tabletop.
The bearing column can also be modeled as a cantilever undergoing bending in both “pitch” and “roll” directions, since the point load is placed at a corner. The actual cross-section of this member would depend on what bearing design I use. Here, I am assuming a solid rectangular cross-section to build the spreadsheet. The correct second moment of area can be substituted once I nail down a bearing design.
The crossbeam undergoes torsion and bending simultaneously. The torsional component is relatively self-explanatory, but I took a while to realize that the corner loading meant that one leg would be (at least partially) unweighted, allowing half of the crossbeam to behave like a cantilever. This first-order model neglects the dead load from structural components, leading to a relatively large contribution by this term. The symmetrical design means that the weight of the structural members will partially cancel out this contribution.
The legs act as cantilevers under bending. However, after realizing that the crossbeam bends and transfers load differentially between the two legs, I became a bit confused about how to formulate the deflection contribution from these members properly. I plan to sleep on it and revisit in the near future.
The total load-induced error from the structure alone was about 5 mm, which is approximately twice as high as I have apportioned. On the bright side, the vast majority of this error came from the tabletop. My current model assumes a flat sheet of plywood without any bracing, trusses, or composite panels. There is a great deal of room for improvement in that area at little cost, so my focus for next week will be improving the stiffness of the tabletop module.
The geometric error, which in this case comes entirely from the bearing, is around 2 mm. This comes from assuming a 0.25 mm radial clearance in plain sliding bearings. As we discussed in class, appropriate use of preload can all but eliminate this sort of error, so I am not excessively worried about this either.