Since the structure is statically indeterminate, we need to get the stiffnesses about right in order to get the answers about right. They don't need to be precise - apart from anything else, I'm not analysing a particular wheel, but rather a typical case. Since real wheels vary from one to another, depending on which of the myriad available rims, spokes, tyres etc. are used, there's some leeway - but they do need to be 'about right'.
My parameters were guessed. I took what I knew about bicycle wheels and typical material parameters and guessed some values. Since then, I've found some other references to FE analysis of wheels, one of which has a detailed derivation of material properties, so it's useful to compare them. The other documents (both pdfs) are:
|Parameter||My value||H.P. Gavin's value||A.D. Hartz's value|
|600.0mm diameter circle, the plane of the rim being central between the flanges.||619.0mm diameter, but dished (since modelling a rear wheel)||618.8mm diameter|
|Hub ends of the spokes on circles of 40.0 mm diameter. The two hub end circles 50 mm either side of the plane of the rim.||44.4 mm diameter, left flange 36.7 mm from plane of rim, right flange 14.1 mm.||36.0 mm diameter|
|2 mm diameter axial load only members. Hence, a cross-sectional area of 3.14 mm2.||1.83 mm diameter, so area 2.63 mm2||unknown (reported as area 62.34, but I don't believe that)|
|Young's modulus 205 000 N/mm2, this is a typical value for steel. Since they are axial force only, no other parameters are significant||Young's modulus 206 000 N/mm2||Young's modulus 210 000 N/mm2|
|A rectangular hollow box of external dims 25x10 mm, side-walls 2 mm thick, top and bottom faces 1 mm thick. Hence, cross-sectional area of 82 mm2, an I value in the plane of the wheel of 1187 mm4, and an I out of the plane of the wheel of 6847 mm4.||
I in plane 795 mm4,
I out of plane 1200 mm4.
A 138.4 mm2,
I in plane 1469 mm4,
I out of plane 9399 mm4.
|Young's modulus E=75 000 N/mm2, poisson=0.3, implying G=28 850 N/mm2. These are typical values for generic aluminium alloy.||
E=69 000 N/mm2,
G=26 000 N/mm2
E=70 000 N/mm2,
neither poisson nor G specified.
So my wheel has a slightly (3%) smaller diameter than either of the other analyses, but it's still larger than a 26" wheel, so I'm reasonably happy with 600mm as 'about right'.
My hub geometry is almost the average of that used in the other two analyses, so that's probably reasonable (all three of us probably aren't badly wrong...).
Spoke section and material properties are similar - I have slightly thicker spokes (about 20% more area), but the Gavin paper discusses alternative spokes, and 2mm diameter spokes are part of that discussion, so here too my analysis ties up with other published work.
Rim parameters are more interesting. This is particularly so because the Gavin paper has an interesting section on the derivation of the flexural values, which were obtained form a reasonably complex experimental examination rather than geometrical derivation. This was so that the effect of eyelets and so on in the rim were properly accounted for. However, for the most significant value - bending in the plane of the rim (ie, about an axis parallel to the wheel axle), my value falls between those of the other two analyses (in fact, within 5% of the average of the others). If I were doing the analysis again I'd probably do a slightly thinner rim, maybe 20mm rather than 25. That brings the values down to A=72mm2, Izz=984mm4 and Irr=3936mm4, which are closer to the Gavin figures. However, one of the complaints I've had about the analysis is that it doesn't reflect more modern more stiff rims, so actually, to find I have modelled a rim that's 50% stiffer than the one good set of experimental data on rim stiffness goes some way to discrediting that complaint.
I also have a slightly stiffer than the other analyses material for the rim. Combining I values and material stiffness, my rim is 62% more resistant to bending than the Gavin analysis rim.
All in all, I'm agreeably pleased with the correlation between my data and that used by other analyses of bicycle wheels.
The one issue
where my figures are arguably adrift is rim stiffness. However, the only
complaint' I've had about this aspect of my model is that my rim
isn't stiff enough, yet here it's 60% stiffer than the results of a
detailed experimental examination. As such, I think my figures remain
a reasonable compromise.
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