Wheel Design and Structural Analysis Program

The use of the analytical wheel model for evaluating structural properties of a wheel design is an obvious application that may have more uses than the truing algorithm.

The goal of this application  is to evaluate stiffness in various directions, buckling spoke tension, maximum and minimum tension in the spokes, and maximum deflection under normal and limiting case loads.  The wheel can be analyzed with different combinations of tension, spoke diameters, spoke lacing patterns, hubs, and rims.  The wheel is an interconnected structure in which all components and loads affect one another.  Human intuition about the structural mechanics fails us in figuring out how one change will affect the wheel as a whole.  The resulting myth and false information about what different designs achieve for the rider leads to poor designs.  This tool allows some quantitative measures of wheel's properties.  Exploration of the parameter space replaces intuition with real science.

Some changes have to be made to add the capability for external loading to the wheel.  We have added point loads representing externally applied radial, axial and azimuthal loads.  Combinations of these loads can simulate virtually any riding condition.  The radial load represents the vertical load of the wheel against the road.  The axial load is a side load from turning or rocking the wheel.  Azimuthal load is the load from pedaling or braking.  The model assumes mechanical equilibrium and computes the reaction loads for the externally applied loads.

We have designed the calculation so that a number of values can be entered for a parameter or load so that the outputs may be plotted as a function of the range of inputs.

In its present form, the code exists as a Matlab program.  I have not uploaded it to the Mathworks because I still hope to commercialize it.  Proposals on creating an app are welcome.

Will the real Thomas Wilson Please Stand Up

I googled Thomas Wilson bicycle wheel.  Interestingly, I personally was not on the first, second, or third page.  I did find a Thomas Wilson who is a service manager at a cyling shop, a storyteller who wrote about his 6 year old learning to ride a bicycle and his brother 20 months younger who duplicated the feat almost immediately.  I found that Thomas Wilson was in Back to the Future as the redoubtable, Biff Tannen.  I have long know about Tom Wilson and Tom Wilson Jr. , father and son, who have drawn the cartoon Ziggy, for 45 years.  Tom, Tommy or Thomas Wilson has played several times in the NFL, NHL , MLB, and  professionally for the Australian national team.

There are innumerable Dr. Thomas Wilson’s.  You could be treated multiple times for every conceivable ailment all by one or another Dr. Wilson.

There are professors, lawyers, and all sorts of other professionals who appear when googling the name.

Thomas Wilson, it is like not having a name at all.

Truing Based on Measured Influence Functions

The truing algorithm is a model-based control algorithm.  The "model" is a structural model of the wheel that predicts how much tension and displacement change when a spoke nipple is adjusted.  I have worked for a number of years on an analytical solution to the structural mechanics model of the bicycle wheel.  Another approach is to measure experimentally the structural response instead of calculating it.  In either approach, the model of the wheel boils down to a relationship called the influence function.  An influence function is the response of all the measured variables to a unit input applied to one controlled variable.   In the case of the wheel, the controlled variable is the spoke nipple rotation.  The ensemble of influence functions from all controlled variables is the influence matrix.   Because the wheel is very closely approximated as a linearly elastic structure, the influence functions can be combined by superposition.  The predicted result for set of spoke adjustments is just  a matrix multiplication.  

The influence function is something that can be measured by simply turning each spoke nipple a single turn and measuring how much the axial and radial displacments and the tension changes all around the wheel.  We have measured the influence function since the very beginning of the project to to validate the calculated influence function as a way of testing of the analytical modeling.   Obviously, we could just use the measured influence functions in place of the calculated in the truing function.  The trade off is the accuracy of the measured functions versus the accuracy of the calculated model, the time and difficulty of making the measurements versus gathering the modeling data.   To some people, the measured just seem simpler to understand.  For different users, one or the other might be favored.  So I have created a version of the truing algorithm that uses measured influence functions.

It is not necessary to measure the influence function for every spoke.  A bicycle wheel typically has a group of spokes that are geometrically similar.   The similarity is that they cover all the combinations of leading/trailing, drive/nondrive, inside/outside flange positions of the spokes.   This is the repeatable group.  The influence matrix can be formed by repeating and shifting circularly the influence functions of the repeatable group around the rim.  Matlab has a function circshift that facilitates the full matrix formulation from the repeatable group.  For a typical wheel that has crossed spokes on both sides the repeatable group would consist of four spokes.   For a radially spoked wheel, the repeatable group is 2.   A wheel that is radially spoked on one side and crossed on the other could have a group of 3 if the radial side does not have an inside/outside attribute.

The influence matrix is the modeling input to the control problem to bring the wheel from the as-found condition to a true, round equally tensioned condition by set of spoke adjustments.  The control problem is overdetermined.  That is the number of variables that one would like to control to zero error (displacements radially and axially and the spoke tensions form set of errors that are three times the number of spokes).  Consequently, we have to solve a problem minimizes some combination of errors.  In our case we form a quadratic summation of squares of errors and proposed adjustments of the spokes which we call the cost function.  This function is easily minimized by solving a linear system of equations to yield a gain matrix. The adjustments to true a wheel is the product of the gain matrix times the set of error vector.

If you are interested in the measured influence function version of the code, drop me a comment and we will figure out how get it to you.  If you would like to see some equations or results posted here, you can mention that in the comments and I will put some effort into doing that as well

TW

Hasty Post on Modeling Equations

My last post on Introduction to Introduction to the Model Derivation and Overview of Equations was a bit hasty.  After finding a way to link documents that actually transferred the equations, I was eager to try it.  However, there are still some problems with the linked document.  The equation numbers have been stripped so there are gaps in the text where the reference is given and there are no numbers attached to any equation.  The other problem is that I was actually in the midst of an update on this version to add external load.  I had not gotten very far but the incomplete editing would probably be confusing.  The equations are difficult enough to understand without the inconsistencies inflicted by the author.

Please give me your forbearance.  I will update the document as I have time over then next few days. In the meantime, some comments from readers would be appreciated.

Introduction to the Model Derivation and Overview of Equations

In this linked document, the overall layout of modeling equations is described.  The matrix form of each equation is given.  Detailed derivations of the matrix elements will be given in subsequent posts.

It appears that the equation numbers generated by Mathtype are not recognized by Googledocs and are not given.  This may make the document unusable. Please advise if you cannot follow the text because of this problem.

Overview of Matrix Structure of Wheel Model

Background on Bicycle Wheel Modeling and Wheel Truing

In this post, I have uploaded my background section from my derivation of the truing algorithm.  Please comment if the link works for you and you can read the document.

Background on the Bicycle Wheel Model and Truing Algorithm

Reintroduction

In the linked document below, I have updated the introduction to the bicycle wheel.  This post is a small experiment.  If it works, I will publish my wheel derivation in sections.  Let me know in the comments if you are able to go to the link and read the document.  There are a couple of equation objects that are almost correctly formatted.  The only problem is that the symbol appears above the line rather than on the line as intended.  

Introduction