The single-column design is made up of 4 structural elements:
- Bearing column
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.