The status of the wheel problem

For those who are checking this blog daily, here is the news.  I have not stopped working on the wheel problem but I sort of gave up on the math blog.  Let me offer two reasons.  Blogger is not friendly to equations and I cannot write a math blog without equations.  It is not impossible to include equations in Blogger but it is awkward.  And the second reason, no one is apparently reading the blog so there is little incentive to overcome Blogger's mathematical limitations.  I continue to work on the writeup in Word.  Word is a familiar writing environment for me and, with Mathtype, it is relatively friendly to equations and figures.  If you are interested in seeing the current state of the solution in equation form, leave me a comment and we will work something out.  From now on, I shall just talk about the algorithm in general terms without equations.  Some people may be more comfortable with this approach anyway.

The current state of the wheel model is that I found by comparison to experimental data that the model does not predict actual wheels very closely on the single spoke perturbation test.  Also, as the tension in the spokes increases, the comparison gets worse.  As a first guess, I have added hoop stress to the model following the general approach for including axial buckling in an Euler-Bernoulli beam model to the wheel model.  The result is that the hoop stress term generally improves the axial comparison but has almost no effect on the radial results.  I find this puzzling but have reviewed the equations to the point that I think that they are right.  Right in the sense that they a correctly implement the hoop stress that I intended into the axial and radial displacement.  The problem is not a programming error or error in the algebra but is the actual consequence of the what I set out to do.  So I am looking for a new direction.  I think the problem lies instead with the fundamental equations.  I am current working to add cross-bending stiffnesss to the wheel's stiffness matrix.  I don't have a compelling physical reason for this term.  I argued that it was zero in the original derivation because the rim is a slender beam and that the spokes are thus must be attached fairly near the centroid of bending.  The cross bending stiffness term arises when force is not applied at the centroid.  With a symmetric wheel (symmetric about z=0), these cross bending terms are mathematically zero.  I can argue now that perhaps the spokes are not at the bending centroid and thus the term should be considered.  I have not performed an analysis of the rim to see if this idea has physical merit.  I just observed that the shape of the radial  deflection mirrors the shape of the axial deflection and that the cross-bending stiffness would give a mechanism for that dependence to appear proportionately in the radial bending equation.   I have the algebra roughed out for the solution to this model but have not put it into the Word document.  I have also not programmed the new equations.  The algebra is getting more and more tedious which is the subject of my next post.  I am still making sure it is right.