Deterministic Design of Simple Linear Motion Axis

Overview

FRDPARRC table: linear motion axis design

I plan to ultimately use this linear motion axis design in my standing desk, so the design here will be driven by broader design requirements I have in mind for the desk. I have narrowed the concepts I am considering down to some form of sliding-contact bearing, partly to minimize system cost and mechanical complexity, but more importantly to try to exploit the binding characteristics of slider bearings in creating a self-locking axis.

Error Apportionment

As seen in the FRDPARRC table, the maximum deflection I am willing to tolerate at the extreme corners of the worktop is 0.5″. This is a perceptible amount of movement, but not uncommon in portable furniture. While the first-order analysis we did in class simply summed the error contributions from bearing slop and worktop flexural deflection, I decided to distinguish between the two sources of error.

The error motion due to radial clearance in the slider bearing takes place against negligible stiffness — i.e., a tiny load applied by the user in a new direction can almost instantaneously cause the system to take up all the backlash, going “clunk” in the process. On the other hand, the load-induced deflection is resisted by the flexural stiffness of the worktop structure. There is a more or less linear mapping between load applied by the user and deflection. In my experience, this form of motion has a far smaller impact on user experience since small amounts of flexing under load are expected by most people, while the sudden “backlash” type motion creates an unnerving sense of instability. To account for this different weighting from a usability standpoint, I assigned a 80:20 split between load-induced and geometric errors.

My error apportionment spreadsheet can be viewed here.

Self-Locking Axis Analysis

One of the most important functional requirements for my standing desk is that the mechanism is self-locking, even when the drive system is de-energized. This is critical for safe and efficient operation of the system. In a screw-driven system, one way to achieve self-locking is to increase friction in the threads. The cost of increasing friction (by using a tighter thread fit class, oversizing balls for ballscrews etc.) is that a more powerful and more expensive motor is required to overcome the friction whenever the system is actuated. An alternative (reciprocity!) way to achieve self-locking is to reduce the torque required to support the design loads by specifying a leadscrew with a finer pitch. Unfortunately, there is also a tradeoff here with actuation speed, since a finer-pitched leadscrew will have to rotate through more revolutions to move the carriage by a given amount.

I have decided to focus my efforts this week on utilizing the usually undesirable binding behavior of sliding linear bearings to produce a linear motion axis with self-locking capabilities. Intuitively, the jamming of sliding bearings is associated with rotation caused by loads acting away from the center-of-friction, which means that it can be “switched” on an off by an applied moment. Since the linear motion axis will most likely be placed along an edge of the tabletop (as opposed to through the middle), the dead load associated with the tabletop structure and things piled on top is inherently off-center. There is a chance, then, that this concept will work. The above pages from my notebook contain my analysis to verify this intuition. From this analysis, it appears that it is workable. My next step is to consolidate this analysis in a spreadsheet and try out several what-if scenarios to determine whether (and if so, by how much?) it improves upon a conventional self-locking design that relies on friction torque in the leadscrew.

Three-Groove Kinematic Coupling Fabrication

My original plan was to machine my coupling from plywood and glue in steel contact elements. Unfortunately, the CNC router in the Makerworkshop is down for repairs. Inspired by Prof. Slocum’s demonstration using bagels and fruits, I decided to make a version of my design using soft materials. I thought this would be sufficient to help me build intuition. I had already acquired all the materials and generated toolpaths for the original plywood and steel design, so I plan to also produce that once the machine is back up. A performance comparison between my soft materials mockup and the final version would be interesting.

Melon Coupling Fabrication

I had a perfectly ripe honeydew melon I was looking forward to eat, and decided to borrow a small part of it to build my mockup. I cut the melon into 2 circular discs and punched holes in one of them, placing each hole on a vertex of an equilateral triangle. I then “transferred punch” those holes onto the other disc using a chopstick, thereby laying out the positions of the matching vee grooves. Cutting the vee grooves was a good chance for me to practice my knife skills.

I had hoped to glue the bearing balls to the melon disc over the holes. I had even whipped up a batch of starch-based glue to try this (gelatinizing cornstarch is a popular cooking technique in Chinese cuisine!). Unfortunately, the moist surface of the melon didn’t take well to adhesives, and I had to fall back on making the top half of the coupling from cardboard, to which I glued the steel balls.

Repeatability Testing

Melon Kinematic Coupling Repeatability Test from Shien Yang Lee on Vimeo (CC-BY-SA 4.0)

I tested the cardboard-melon coupling for angular repeatability using a laser pointer projecting onto a wall 59-inches away. The maximum spread over 5 engage-disengage cycles was 5/8″. Converting this sine error to an angular error using simple trigonometry, we find that the coupling is repeatable to within 0.607°. Extremely impressive for a coupling made out of ripe (so soft!) melon and cardboard with minimal measuring. I think this demonstrates how robust the concept of exact constraint design is against fabrication and material inconsistencies.

Melon coupling repeatability test calculation

Machined Plywood + Steel Coupling

Update: See the fabrication and testing of my revised (non-melon) kinematic coupling here.

Three Groove Kinematic Coupling Design

Concept Selection

One of the tasks for the course this week is to design and build a three groove kinematic coupling. You can read more about kinematic couplings here, but they are primarily used to repeatably position objects relative to each other. I don’t yet have a specific application in mind for this coupling, apart from a vague notion that I would like to eventually use it to fixture some sort of camera in a future project. Therefore, I will be building my coupling to a 2.5″ coupling circle diameter — a reasonable size to accommodate most hand-held cameras.

In the interest of controlling cost, I have chosen to use plywood for the top and bottom plates of the coupling. To select the material for ball and groove contact elements, I estimated the Hertzian contact stresses resulting from anticipated loads (from a ~1 kg camera) using Prof. Slocum’s design spreadsheet and found that a wood-on-wood interface would have been feasible. However, I ultimately decided to use steel for the contact elements in order to avoid potential issues that may result from material and geometric irregularities found in biological materials. Additionally, gluing in steel balls and dowel pins would be easier and faster than machining them out of wood, which would require either a time-consuming workholding setup or a CNC machine.

Analysis

I adapted the template spreadsheet to predict the stiffness and error motions specific to my design. You can view the full spreadsheet here. Some of the key insights from this analysis are:

  • With a preload of 1.48 N per ball, as would be the best case scenario with the 3 lb max.-pull magnets I am intending to use, and a 2-kg object (reasonably large SLR or medium format camera) on the coupling, the maximal lateral force it can tolerate is approximately 13 N when applied in line with a vee groove. Lateral loads above this threshold would cause the coupling to come unseated.
  • An RMS stiffness of 62 N/micron. This means that the Z-position should be capable of achieving sub-micron repeatability in the face of minor variations in the preload or weight of the upper piece of the coupling. This is important as it would facilitate interchangeability of the part being fixtured.

Detailed Design and CAD

After convincing myself of the feasibility of my design, I built a solid model to check that I haven’t made any mistakes in my geometric calculations and to generate toolpaths for the CNC router I plan to use to machine the plywood boards that will receive the steel contact elements. I also decided to use a pair of neodymium magnets inset into the plywood plates to provide a more consistent preload for the coupling. These magnets will be glued into circular pockets positioned at the center of each half of the coupling. My CAD models can be downloaded here.