It took me a while to uncover some of my older notes; I apologize for not replying sooner. Thanks for your reply to my first post; comments regarding your three points:
1) I don’t understand the distinction you make between, “…how much force does it take to displace the spokes vs. displacing the rim?” as when one uses the term “wheel stiffness” it’s the whole system—as you mentioned, but then what’s the difference? Without knowing the parameters you used in your FEM analysis it’s difficult for me to comment on your modeling, but I still assert that the rim can—and usually does—make a bigger difference to overall wheel stiffness than the spoke bracing angle for the typical ranges encountered with (a) rim section depths vs. (b) hub bracing angles. Sure, I understand it’s a hub review (!) but this is an important issue to keep in context.
2) I must be missing something here. As I said previously, the lateral force at the rim is directly proportional to the displacement for a single spoke; that force being equal to sin(bracing angle)—call it simply “sin(a).” Let’s assume a different hub with bracing angle b. Then the (lateral) force at the rim is proportional to sin(b) and the ratio between the two cases –what we’re interested in for wheel design purposes–is (sin(a))/(sin(b))=a/b for small angles, where the “=” sign is understood to be approximate. That appears to be linear to me. For a symmetrical (e.g., non-disc front) hub with the same bracing angle on each side, you could call the ratio proportional to 2a/b since there’s an equal contribution (with a pre-stressed structure, that is) due to the spokes on the other side, but it’s still not a quadratic relationship. This simple analysis also agrees with the graphs produced from FEM analysis done by Mark Rodamaker, “Design and Analysis of an Optimum Bicycle Racing Wheel,” Finite Element News, Feb. 1989, pp. 33-37. Figures 4 and 8 show that the lateral displacement is linear with force up to the point where some of the spokes go slack.
I also found the paper I referred to in my first post. The citation is “Bicycle-Wheel Spoke Patterns and Spoke Fatigue,” Henri P. Gavin, Journal of Engineering Mechanics, August 1996, pp. 736-742. Gavin’s work is analytical, not FEM—my error—and verified experimentally. It was the discovery of Gavin’s paper that led me to work on deep-section wheels in 1998, where I performed a series of wheel stiffness tests on various wheels (15 in all; 5 rear and 10 front wheels) including several intended for tandems as part of the development process for an early entrant in the “deep section rim with low spoke count” sweepstakes. The most fascinating revelation (to me, at least) was Figs. 5 and 6, which show curves for radial and lateral wheel stiffness vs. rim bending moment of inertia about the axis parallel to the axle—in other words, these are graphs for radial and lateral stiffness vs. in-plane (or radial, if you like) rim stiffness. The key takeaway is that the wheel’s lateral stiffness increases just as fast as the radial stiffness for an increase in radial rim stiffness. Hmmm…
This is (or was) the result: http://www.sandsmachine.com/a_bil_t8.htm. The lateral stiffness of these 24 spoke wheels was nearly the same (within 10%) as a pair of 48 spoke wheels I tested that came off of a Santana Sovereign. Interestingly, the Phil rear hub I used for my wheels was a 145 mm version (flange spacing of 55 mm), whereas the Santana rear had a 160 mm hub with 60 mm flange spacing and the same Wheelsmith DH-13 2.3/2.0 mm single-butted spokes as I used. The difference, clearly, is the rim: 40 mm deep FIR Rialto for my wheels vs. 19.5 mm deep Mavic T217 (a popular box-section tandem rim at the time) on the Santana wheels. The rim widths are close, with the Mavic being slightly wider (22 mm) than the FIR rim (20 mm) I used. Even I was amazed at how much of a difference the radial rim stiffness makes in lateral wheel stiffness. To reiterate, the difference in radial rim stiffness is approximately (40/19.5)^3=8.6 times. The fact that there are only half as many spokes along with a lower spoke bracing angle yet the lateral stiffness is comparable indicates that the rim is the dominant effect, not the bracing angle. A wider rim and/or more spokes and/or a wider hub would’ve made these wheels even stiffer laterally, but the FIR rims only came in 16 and 24 holes (I also have a 16 spoke front we’ve used on our tandem and even occasionally on the triplet), that’s how I designed the Bilenky (i.e., 145 mm rear), and they’ve served us well for 15,000 miles of riding to-date, so they’re obviously stiff enough as well as strong and durable enough: we’ve never broken a spoke.
3) I would agree with this assessment.
Feel free to contact me offline if you’d like copies of the articles mentioned.
Thanks for listening,