Problem resolved?

I think I have figured out my flaky wheel results.  I was comparing experimental data which I took using an old wheel against the model.  It was a different wheel than the wheel I had used for my main test cases.  The wheel did not seem that different so I expected things would be similar, but the results were really bizarre.  In the end, I found a couple of programming errors and I learned something about the model.   The errors were things that were perfectly obvious on inspection, but if your results look good it is hard to make yourself inspect carefully.  It takes something really strange to focus the attention on the programming and really see anything wrong.

The thing I learned about is the effect of exceeding the buckling load in the model.  Buckling limit is one of the things I wanted to be able to calculate with the model.  I know that the spokes increase the buckling load compared to the bare rim but I did not know how to calculate from the model.  I am on the track of a tidying up the theory.  Just by chance, the stiffness data (Izz, Irr, and J) and spoke tension I picked for this wheel were right on the limit for buckling.  The matrices are singular at the buckling point so the inverse of the matrix near it is unreliable.  That accounts for the flaky results.

The good news is that putting stiffness parameters from my first case (a good bit stiffer to both bending and torsion) pushes the buckling limit out to higher value.  With that set of parameters, the model gives excellent displacement results in comparison to the experiment.  In playing with the parameters, the predictions of displacement and spoke tension are very sensitive to stiffness parameters near buckling but not so sensitive away from it.  The stiffness parameters I first estimated for the second wheel are not particularly accurate.  I do think this wheel is close to the buckling limit.

I am working on getting some help to calculate stiffness parameters accurately for a given rim profile.

Some progress ....

I woke in the middle of the night last night with a thought of a possible error in the wheel model.  I lay awake thinking about it until 6 am then got up and, even before coffee, checked the code to see that sure enough that it was indeed a problem with the model.  I fixed it, ran some tests, and could see that things were better but still not completely correct.  I am using a simple test of logic, a symmetric wheel should have symmetric elements of its matrices.  A leading spoke should be same magnitude but opposite sign as a trailing spoke.  So far today, I found three errors.  Middle of the night wake-up error was the biggest, the next two smaller, and yet the modeling matrices are not exactly symmetric as I expect they should be. It is surprising but true that it is easy to see asymmetry but it is hard to detect where it comes from.

Perhaps, I should sleep on it and the answer will come to me in the middle of the night. For all those following along at home, Janice thinks that the wheel problem can wait and the answer will come to me as I complete the kitchen cabinet additions.  I shall follow her sage advice.

One step forward, Two steps back

Well, I learned today that all is not well in my model.  I tried to apply the model with rim profile twist to an set of data that I took on a Real Design Supersphere wheel.  The original model without twist or buckling did reasonably on this wheel.  That model predicted the response to the single spoke perturbations and the truing algorithm trued up the wheel up nicely (>.15 mm in both directions).  The wheel has fewer spokes and lower tension than the last wheel I reported that did very well with the model with twist.   It seemed likely that this wheel would be have less effect from the buckling instability because tension was lower.   I thought success was a foregone conclusion.  Not so fast.  

The short version of the story is that the new model of this wheel is unstable at a much lower tension than actually exists in the wheel.  It becomes stable if I decrease the spoke tension or increase torsional stiffness beyond reasonable values.  In further investigation, I also find that a symmetry that the wheel should possess (every forth spoke should have the same influence function) turns out not to be true for the new model.  I think these are two separate problems, but maybe not.

I do not have time to work on the wheel right now.  I must make some cabinets.

A summary and conclusion of the bicycle wheel problem

I have reached, if not an final answer to the wheel problem, a stopping point.  I have used the methods of flexural-torsional bending to solve the structural mechanics problem.  The results are pretty good finally.  The predicted and measured displacements are certainly good enough to apply the LQG control theory to the truing problem.   So the part I set out to do is done.  The structural model is a success and there are many more things that could be done with it besides just truing at this point.  What I had hoped was that someone who wanted to commercialize this idea would take it over now.  The most I can offer in the future is a journal article to document the method of solution.  If you are interested in a great structural model of a bicycle wheel or a wheel truing algorithm, I am having a sale this week.  Best prices.  Can't be beat.

Here are some final images of the comparisons of prediction versus measurement.  In this comparison, a single spoke is tightened one full turn and the change in shape is plotted.  I think I nailed it.  The input wheel parameters are dead nominal.  There was no adjustment of parameters to improve the fit.  This is the raw data.

The sample wheel is a difficult wheel to model.  This wheel has a large number of spokes and a fairly high tension.  As a result, the structure is close to the flexural torsional bucking load for the rim.  The taco shape of the rim reflects how sensitive the wheel is to the spoke perturbation.  In the figures, I am also showing the earlier, less successful models that do not account for the instability.  The good comparison from the final model and experimental data  is the turquoise line (final model) and red line (experimental data) in each figure.  I previously got acceptable results on other wheels with fewer spokes or lower tension.  The wheel shown here was the one that was hard to get right.  The final model reduces naturally to the earlier model as the parameters leading to the instability are modified to make the wheel more stable.  The first model had no elastic instability modeling.  The second had flexural instability but not torsional.  The final model had flexural and torsional instability.  This mode gives the lowest buckling load.