Trying to find a way out.

Dear readers,
I had a comment.  First one ever.  Thank you, Anonymous.  Solving math problems when the answer is not in the back of the book is sort of like walking a tight rope.  You never know if you will fall.  The act of balancing is similar for all who choose to attempt the solution.

Not much is happening on the wheel problem.  I am more committed to being ready for a bike tour in Italy than in math these days.  For readers who do not know me in real life, I am going on a bike tour in the Dolomites in June.  I have been riding 3 or so hours a day to be sure to be the next to slowest cyclist on the tour.   No Lanterne Rouge for the Geezer.  The time for training cuts into the time to do wheel research.

I have been planning to apply the wheel algorithm to build wheels.  The first candidate for building may be the wheels for my old classic Raleigh Grand Prix.  I broke a spoke on the old Mavic wheel that came with this bike (it was not the standard wheel) trying to test the wheel model.  I think I should build a wheel to replace it.   The original wheels were Mavic tubular rims on the very classic, high flange Normandy hubs.  The Normandy hubs were the only quality component.  The rim was a low tech aluminum box.  The spokes were extremely low tech galvanized steel spokes.  So, I plan to build a 36 spoke, three cross clincher wheel set around the original Normandy hubs using a new clincher rim and spokes.  Dear readers, what would you recommend as a rim and spoke combination?   The Raleigh Grand Prix was a touring bike with carbon steel frame and 2x5 gearing.   I have all the original parts.  All parts are functional.

What say the Internet readers of the  Solving Math Problems blog?  What modern rim and spokes should I pair to test the wheel algorithm and build an interesting and useful wheel set for a classic bike?

TW

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.

Hoop stress and cross-bending stiffness for the bicycle rim

I have been revising the structural model of the bicycle wheel.  The revised bicycle rim model has added the effects of hoop stress and cross-bending stiffness to the rim structural model.  The hoop stress is the stress compressing rim along its circumference in the azimuthal direction due to spoke tension.  Cross bending is the phenomenon that couples a bending force in one direction with a displacement normal to it.  The term is computed by using an off-diagonal coefficient in the stiffness matrix. This coefficient is called cross-bending stiffness.   A bending moment directed in the radial direction causes bending not only in the radial direction but also causes bending in the axial direction and vice versa.   The cross bending stiffness is the proportionality constant in the stiffness matrix that relates the axial bending moment to the radial curvature.  Both hoop stress and cross bending stiffness were previously neglected as small (and inconveniently tedious to include in the derivation).

The motivation to revise the model was the poor comparison we observed between some measured data for the rim model and the model predictions.  This revision has been underway since I discovered the poor comparison last April, eight months ago, more or less.  I suspected that hoop stress was part of the problem because the wheel that showed the poorest comparison was a cyclocross wheel with high tension and 32 spokes.  The number of spokes and the high tension combine to give a much higher hoop stress than other wheels I had tested that compared better.  Also the shape of the measured wheel with a single perturbed spoke (one spoke is tightened one full turn) was deformed in the familiar taco shape of a damaged wheel.  The taco shape is widely described as being a consequence of buckling of the rim.  Bucking is an elastic instability associated with the rim in compression.  Hence, I supposed that the physics related to the buckling was also related to the shape observed in the perturbation experiments.

The changes have been added one at a time.  First, only the hoop stress was added to the axial bending equation.  This term helped the axial displacement show the expected taco shape but with a smaller amptitude than the experiment.  Better comparison was obtained by various parameter changes that resulted in less stiff rim or more stiff spokes.  I expected similar improvement for the radial displacement.  When I added hoop compression terms to that equation, it hardly affected the radial displacement at all.  I sought out some additional physical phenomenon that had been neglected and found that potentially the cross-bending coupling might act in the way needed to improve the comparison.  I started making this addition in early December 2014 and had time during Christmas travels to work on it.   The solution came together while we were in California.  I finished the derivation in January and programmed it shortly afterward.  I also have written a separate document in which the equations are derived.  This document improves on the original derivation not only in adding the hoop stress and cross-bending terms but also because the equilibrium force balances are derived with new illustrations that aid in understanding the directions of force vectors in the differential element of the circular rim.    

The derivation is about 40 pages long at present count.  The programming of the new model is only about ten lines of coding.  It is an amazing compression of the time and thought, the physics and mathematics of the derivation into a compact algorithm.  The compression is aided by the tools for matrix algebra provided in Scilab but even a Fortran program would probably only be 20 or 30 lines.  The compactness of the final answer is beneficial in many ways, not the least of which is that there are not so many signs and matching parentheses to go wrong.  The symmetry of the solutions for the radial and axial equations is reassuring if not scientifically sound evidence of its correctness.  

The revised algorithm for the coefficient matrices is quite a bit shorter than the previous wheel models.  This is not because the model is simpler, it is not.  It is because I had some new insight into the matrix equations.  I learned that I could construct the coefficient matrices because they possessed the special symmetry of Toeplitz matrices.  This form allows the matrices to be defined by the elements of the first row and column.  I also was able to reduce the computations to high level matrix expressions, eliminating some intermediate variables and some complicated term-by-term computations involving individual matrix elements contained within for loops.   The end result is that the model programming and the equations in the derivation document are very close to the same.

The results are less successful in matching the experimental results than hoped.   The following three figures show the orginal model, the revised model with cross-bending stiffness, and the experimental results.

The tension is a pretty good match which suggests that something is right about the model.  The axial shape has the right general shape but the wrong magnitude.  The radial shape seems totally wrong.  It is very interesting that the experimental radial displacement echoes the two hump (inverted dromedary?) shape of the axial displacement but the computed displacement does not show any evidence of the taco shape.  The stability of the spoke model in tension totally eliminates the instability of rim bending in the radial direction (the instability that would be present in the bare rim unsupported by spokes).  The plots shown are for nominal dimensions and material properties, but I experimented with a range of parameter values for stiffness of the rim and spokes.  Nothing moves radial displacement into a taco shape.  

To explain the discrepancy, it occurs to me that I have plotted the displacement of the center of bending of the rim.  That is, Euler-Bernoulli approximations reduce the three dimensional rim to an equivalent one dimensional infinitesimal wire whose bending stiffness parameters are the same as the actual rim.  We are plotting the displacement of that infinitesimal wire.  This approximation neglects the fact that the actual rim rotates about the center of bending and the measurements taken on the surfaces of the actual rim would be affected by that rotation.  The effect on the illustrated results would be an enhancement of the displacement at the center of bending.  A rough calculation assuming the rim cross section is rigid (the effect of the Euler-Bernoulli equations) and using the angle of the rim displacement and the external dimensions of the rim to a displacement due to the rotation indicates that most of the observed discrepancy between measured and calculated would be corrected.

I am setting the bicycle wheel modeling work aside for a while as I work on getting our house in Oak Ridge ready to sell.  Please comment on the results if you are interested in helping sort out the deficiency or if you are just dying for me to do it to see how it turns out.

More on the paroxysmal aspect of BPPV

The good news is that my vertigo has mostly gone away.  Mostly meaning that I am a bit unsteady just when I get up in the morning.  The rest of the day, I seem reasonably steady.  (I can stand on one foot with my eyes closed and touch my nose with one hand then the other.)  I am now focused on preventing the next paroxysm from occurring.

I am still reading online articles about the symptoms, causes, and cures of various types of vertigo.  Of all the possible causes, the rocks in the ear (BPPV) and Meniere's disease (loss of level control in the inner ear) seem to be the most likely causes.  Of the two, you would much rather have BPPV.   Meniere's disease doesn't go away and the vertigo is accompanied by ringing in the ears and hearing loss.  I have some ringing which existed before my recent episode and some hearing loss which also pre-existed.  I think the descriptions for Meniere's are much more severe forms of tinnitus and hearing loss than what I have.  So, I am going to assume the ER doctor's diagnosis of BPPV is correct.

 I really can't find much help online on preventing recurrence.  I have read some websites which advocate homeopathic remedies that are basically a healthy lifestyle.  I was already doing as that as well as I can.  The only thing I read that I might improve is drinking more water.  I am now adding four large glasses (probably 12-16 oz each) per day to my liquid intake.  I think the idea is to change the solubility of the fluid in the ear can so that calcium carbonate is not as likely to precipitate out and form crystals in the inner ear.  My view on extra water is that it can't hurt.  The consequences so far are that I am no longer able to sleep through the night without going to the bathroom or to make the trip from Oak Ridge to Atlanta without stopping.

The next question is whether physical activity, specifically running, biking, or swimming, triggers an onset.  I have read that swimming face down and swinging the head to the side to breathe as in crawl stroke can bring on an episode.  I will hold off on that.  (I haven't swum regularly since 2005 anyway.)  I wonder if the biking in the lowered aerodynamic position  on the bicycle also may cause the otoliths to redistribute into the sensitive areas of the ear canals.  It certainly seemed to be related to my first episode.  I will ride exclusively up on the hoods for now.  I was mostly riding in this position anyway just because it is easier on the back and shoulders.

I am going to ride my bike on a stationary trainer tomorrow to see how it goes.