Bearing Preload System Build and Test

Material Sourcing

This week, I built and tested a bearing preload system for my linear motion axis. As described in my post on the design for this system, the preload is provided by a compliant elastomeric layer in an oversized slider that is compressed by the side rails. Since I only need a small piece of material, I was reluctant to incur the cost of ordering some well-documented engineering elastomer. I took a walk in Blick’s instead in search of cheap materials.

I eventually found some relatively flexible carving blocks intended for making printing blocks of rubber stamps that looked like they might work. They had traditional linoleum pads as well as a softer rubber blocks marketed as being easier to carve. The latter is what I went with as my first-order analysis suggested that excessive preload for the required deformation would be a significant challenge, especially considering my relatively anemic motor.

Build

Using the dimensions calculated by my spreadsheet, I made up a new wooden slider core on the table saw and found a scrap piece of 0.25″ aluminum sheet to use as the bearing pad. On initial dry fitting, I found the slider almost impossible to force into the boxway — probably a result of the very approximate modulus value I used for the undocumented rubber compound. To compensate for this, I took the wooden portion of the slider down very slightly using a belt sander, using a guide to keep the sides square. I also gave the aluminum plate a good brush with some grey Scotchbrite to expose a fresh, smooth bearing surface.

Test

I repeated the “along axis” repeatability test I did on the original motion axis a few weeks ago to try to characterize the effect of adding preload on performance. For a description of the test procedure, see my previous post. I found that the preloaded linear motion axis repeated to within 6 mm measured 2.9 m away. This translates to a total (side to side) angular error of 0.12 degrees, which is more than 50% better than the non-preloaded design. I believe the residual error can be attributed to slight movements of the entire system resulting from motor acceleration (The linear motion axis was just placed on a table without clamping), as well as imperfect alignment relative to the wall (essentially an Abbe offset).

Bearing Preload System Design

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.

Simple design spreadsheet for bearing preload system

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.

Actuated Linear Motion Axis

I ordered a cheap leadscrew assembly intended for low-cost 3D printers from Amazon. The leadscrew came with a brass flanged lead nut, a pair of ball bearings mounted in zinc blocks, and a flexible coupling that fits the NEMA 17 motor output shaft. The leadscrew has a diameter of 8 mm, a pitch of 2 mm, and 4 starts. This means it has an overall lead of 8 mm/rev.

A quick note about running leadscrews directly in bearing bores: conventional engineering wisdom states that it is a bad idea to run threaded rods in bearings because of small load-bearing area results in high stresses. This is still true for leadscrews, but the trapezoidal threadform retains the major diameter across the width of its lands, making it less problematic to run them in bearing bores compared to “standard” fastener threadforms which are triangular and taper to an acute angle.

Repeatability Testing

Actuated Linear Motion Axis Test from Shien Yang Lee on Vimeo.

I tested the repeatability of the actuated linear motion axis in two ways.

Repeatability along motion axis

First, I repeated the “straightness” test I carried out last week for the linear axis without the actuator.After running the carriage back and forth between its extreme positions 8 times, I got a group of laser projection points with the maximum spread of 15 mm at a distance of 2.9 m. This translates to a side-to-side angular error of 0.3°, which is a ~75% improvement from the 1.29° measured on the non-actuated axis. I think there are two factors contributing to this improvement.

First, the motor and leadscrew is able to repeat axial position better than I could by hand. This places the carriage closer to the same locations when each measurement is taken. This theory is supported by the observation that points measured at each position (X0 vs. X100) all lay within 5 mm of each other (angular uncertainty of <0.1°). This suggests that most of the error measured comes from global straightness and parallelism errors in the box way instead of local “wiggling” of the slider.

Second, the leadscrew provides a degree of preload to take up part of the radial clearance. I drilled the slider for the lead nut using a portable drill and did not make the hole perfectly square to the faces of the slider. This slight misalignment places the simply-supported leadscrew (I left it floating in the bearing on the non-driven side) under bending, causing it to act as a preload spring. However, as we learned in class, this is not a good preload configuration since the system’s stiffness varies according to the square of carriage’s distance from one end.

Repeatability orthogonal to motion axis

My second repeatability test was aimed at measuring the precision with which the actuator can move the carriage to a specified position. I attached my laser pointer to the carriage orthogonally, such that it projected a beam perpendicular to the direction of motion. For the adjustable standing desk, this is the sensitive direction.

I recorded the position of the projected beam across the 100 mm-long travel of the carriage on 10-mm intervals. The repeatability at each position was within 1 mm. Note that we are now measuring axial displacement instead of angular error, so using a laser pointer conferred no resolution advantage.

More interestingly, I observed the effects of backlash in this test. I moved the carriage to each position in the following sequence:

0 > 100 > 10 > 90 > 20 > 80 > 30 > 70 > 60 > 40 > 50

Each reversal in direction caused the distance traveled to be short by approximately 1 mm. This is consistent with the perceptible backlash in the low-quality lead nut.

Linear Motion Axis Fabrication and Testing

Fabrication

I fabricated my linear motion axis (boxway) from scrap plywood and a plywood-oriented strand board laminate scrounged from around campus. As mentioned in my previous post, I wanted to keep the slider cross-sectional dimensions to a minimum of 1″ x 1″ in order to accommodate the flange nut when I incorporate the lead screw. Unfortunately, the only sufficiently thick material I could find was the plywood-oriented strand board laminate. This forced me to use the porous and irregular surface of cut oriented strand board as bearing surfaces instead of a smoother material. To compensate for the surface asperities and higher coefficient of friction associated with this material, I increased the radial clearance from my design value of 0.005″ to 0.01″.

After cutting the component pieces to size with the table saw, I glued up the assembly using copier paper (thickness = 0.0035″) as shims to achieve the necessary clearances. For example, a 3-layer stack of paper brings me within 0.0005″ of my design clearances. A mistake I made during this step was neglecting to account for the thickness of the glue layer, this ended up causing my boxway to have excessive radial clearance, increasing error motions.

Testing

Boxway Test from Shien Yang Lee on Vimeo.

I tested the geometric error in my linear motion axis using a laser pointer. I moved the slider between extreme positions on the axis of travel, while applying slight moments to take up the angular “backlash” caused by radial clearance. The position of the projected beam on a surface 4845 mm away was recorded between each adjustment.

Boxway testing: recorded beam positions and analysis

The maximum lateral displacement of the laser beam was 109 mm, which corresponds to an angular error of 1.29°. This is the total side-to-side rotation, which we expect to be twice that predicted by our deterministic geometric error analysis utilizing radial clearances. For my boxway with 0.01″ of radial clearance, I predicted a sine error of 48.5 mm when measured 4845 mm away. The actual value is slightly higher than expected, which I attribute to the mistake I made in not accounting for thicknesses of adhesive layers as well as imperfect clamping during the glue up.

 

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.