Deterministic Design of Simple Linear Motion Axis


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.

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