Buckling of a spoked wheel

I thought my model was pretty close to right so I thought that it should predict a critical spoke stress for buckling or the condition when you turn a nipple and the wheel spontaneously assumes a warped taco shape.  

The method I used was the classic stored energy method.  I calculated the stored energy and tried to take the second derivatives with respect to all the displacements and solve for a maximum.  The result was no maximum, no unstable mode, no critical stress.  It was always stable.  I know this is wrong so the model must be wrong.  This got me to thinking and reading the buckling chapter in Bauchau and Craig and reading a classic book, Timoshenko's Theory of Elastic Stability (that I was able to get off Amazon for something like $50!)  I realized that in my transverse bending models I did not consider the term for longitudinal stress that leads to buckling so I am spending a bit of time putting the azimuthal compression term in the differential equation.  This is going to make a huge difference in the solution.  Timoshenko solves the radial buckling mode but not with an elastic restoring force like the spokes.  I don't know if the revised equation with the axial term will be solvable or not.  The original equations work for low levels of azimuthal stress so it may not affect the truing control problem.

This is still fun.

The control problem is solved

I have implemented a one step control algorithm. AND IT WORKS. This is the simplest version of Model Predictive Control. One step means that given an out of true wheel, the algorithm computes a change in spoke position to bring the wheel back to fully true in one step. The algorithm solves a least squares minimization of a performance index. The performance index for the minimization includes the the tension in each spoke, the axial and radial displacements at each spoke, and the number of turns of the spokes. Including the number of turns in the performance index means that the minimization does not produce an exactly zero error solution of the problem. There is a balance between turning the spokes as little as possible and getting the trueness as close as possible. We say a balance between control effort and control error. Control engineers have found that this approach adds stability to the process in contrast a mathematical approach that tries to solve for an exact match of the setpoints. The control error can be made very small by increasing the weight on the displacement errors with respect to the number of turns. I raised the weight on the errors pretty high to ensure that the one step algorithm would sufficiently close.

I have run several test cases with different out-of-true initial conditions (by entering unequal spoke nipple positions so far). The control solution has solved them all. What I mean is that if I arbitrarily set all the spokes nipple at 10 except for the first one which I set at 15 then compute a set of displacements and tensions in that configuration. I then input that result that to the control algorithm to compute how much to change the spokes to bring it back to true. The algorithm calculates that I need to turn the nipple #1 about -5 turns to get back to true. Perfect. It is a little less because of the inclusion of the control inputs in the performance index.

What this test does is mainly to check my implementation of the control algorithm. It is like when you learned long division in grammar school. After you got an answer, you multiplied your answer by the divisor to see if you got back the number you started with. The division test checks the process of dividing is correct but does nothing to show that the numbers you were working with were right. In the analogy, the wheel model is like the dividend (then number you are dividing into), the computation of the control matrix is like the long division (this analogy is really good here. The computation of the control algorithm is really very much a kind of generalized division.) Checking the number of turns against the amount I dialed in as the error in the spoke initil position is like multiplying the answer and the divisor. The test checks the process by which I computed the control matrix but does not check that the wheel model is right or that the control algorithm is robust to modeling errors. Lets address the robustness first.

Robustness means that the algorithm gets the wheel to true despite a difference between the way the wheel actually responds and the way my model used in the control predicts it. The differences fall into two categories. First is modeling error. Modeling error could be an error in a parameter used in the model, say for example the bending moment of inertia of the rim is off. (I picked this one because I don't have a good way to get this parameter for any given rim. Rim shapes are complicated and the details of the design are hidden inside the rim and I don't have the data nor can I easily measure the data for calculating the bending moments.) It could be an approximation in the math (I linearized the calculation of spoke length for example). It could be something I did not model, like the effect of the spoke hole on the bending stiffness. It could be a clump of glue on the rim that I did not clean off. This class of errors is repeatable and constant in the model. The other type of error is random error. Random error is typically associated with the measurement device. For example, you have to estimate where the dial on the tension meter is pointing between two lines. You can't just read the exact number. The measured value is the sum of the true value plus a random number that has something to do with your ability to see a fractional distance.

Random error could also be some random external influence on the process that we are not measuring or accounting for. Stiction in the spoke nipples is an example of this. By stiction, I mean that when you turn a spoke a small amount the nipple doesn't turn on the threads at first. It just twists the spoke. You go farther and the suddenly the torsional spring force of the spoke is greater than the static friction of the thread and the spoke unwinds into the nipple. There are wheel building techniques to combat this, like turning the nipple past the desired position then backing off just the amount that the spoke is twisted. (Bladed spokes are great for showing the amount of twist in the spoke. Round spokes are harder.) Despite efforts to minimize this effect, the amount of deviation between the number of turns you intended and the actual position of the thread in the spoke is a random variable. The truing algorithm needs to be robust to these errors. Robust means that the wheel converges on true closely enough despite the errors. Failure of the algorithm would be if the errors increase rather than decrease, the errors diminish to a constant value that is not low enough or that the iterations of the wheel algorithm just bounce around randomly but never small enough to be satisfactory. We want the process to be robust. Experience in controls suggest that a process that an iterative process that takes a couple of partial steps to bring the wheel into true is likely to be more robust than the one step routine that I initially tried.

My next step of development will be to test the current control algorithm with modeling errors and random errors. First, I will separate the modeling of the controls from the simulated wheel model so I can incorporate the difference between them. I will create test cases with parametric errors. (A parametric error is a modeling error due to an error in a parameter. I won't be attempting to check for unmodeled effects or approximations errors.) For example, I will double the bending moment of inertia in the control model and try the same out-of-true problems I tried when the model and predictor were the same. I will be comparing the sum of residuals from the cases above where the control and simulated wheel are the same with the cases with the parametric errors.

Next I will try cases that add a random variable to the measurements and to the spoke turns. The idea is to see if the uncertainty in reading the gauges in the truing stand are too large to get the wheel to a true state.

The development after that will be to implement a multi-step control algorithm. I suspect that I will find that I want the control to be more stable so I plan to upgrade to a multi-step model predictive control algorithm. Model predictive control is very easy to implement on this problem. I shall try to put it in.

Testing the model. All the control logic is useless if the model is not sufficiently accurate. To test for modeling errors are two-fold, comparison to other calculations and comparison to experimental data. Jobst Brandt published the total results for cases in which his finite element wheel model is subjected to external loads (vertical, axial and azimuthal loads) for. He also gave the wheel modeling parameters that he used. If his model and my model compare well, it is a pretty strong case that both are right. If they differ then one (or both) of us is wrong. Can't tell for sure. I have a wheel truing stand with dial indicators and tension meter to measure the wheel's variables to gather data for experimental comparisons. So I am working on it. It's fun.

Wheel model is solved

I have got a solvable set of equations for the forward problem. Forward, in my mind, is to find displacements and spoke tension given the spoke nipple position. The reverse problem involves solving for the number of spoke turns to bring the wheel into true and equally tensioned spokes given the current measured displacements and spoke tensions. The reverse problem is the control problem. Getting the reverse problem with the forward in hand is far easier than the forward problem. The forward math problem is just about finished. I need to do a lot of checking of the results before I declare it a really finished. I had some problems getting as many independent equations as unknowns. Many of the equations I could think of were not independent. I had to test for independence numerically because it was not evident from just looking at the equation. I would have expected that the equilibrium constraints in three dimensions of force and six of torque would be independent but they are not, at least not numerically using, the rank test. It is a rainy day so I will be stuck inside. I will be working today on checking the model and doing test calculations. Perhaps some plots of results would be easy enough to load. I have given up on putting equations in the blo

Background and References for the Bicycle Wheel Structural Model

This is an attempt to upload a more complicated document.  A few things did not work.  The reference citation numbers all became # and reference list disappeared. The letters appear crowded together which makes it hard to read.  In one brief spot, the font spacing changes.  It only last from the end of an equation to the end of the paragraph.  No idea how to get the entire document to use the same spacing.   

Hmm. My photos do not show up.  I shall have to work on that.  

I see. I need to upload to the blogger editor.  I can't upload document with photos inserted  Everything takes an extra step.

Bicycle Wheel Truing Algorithm

Background 

The background references on bicycle wheel modeling have been assembled by searching for terms such as circular beams, curved beams, bicycle wheel, spoked wheel, etc. using Google. We also searched for books on structural analysis and linear elastic modeling for the underlying theory of the models. The first part of the background section lists the articles and books specifically on bicycle wheels.  We did not find an appropriate model in this literature, but we did get an approach to develop our own model.  Then, we have a section on the general references on elasticity which we used as background for the equation development of our own model.  We specifically sought general references that discussed stress and strain in cylindrical geometry on which an appropriate wheel model could be based.  Of course, this literature is huge.  The general references list is not exhaustive at all. Once we gathered enough information to formulate a model, we stopped searching.  Other better texts may be out there.  These just happen to be the ones we used.

Bicycle Wheel References

The book by Jobst Brandt, The Bicycle Wheel# is a useful, basic reference on the bicycle wheel. It is divided into three main parts: Theory of the Spoked Wheel, Building and Repairing Wheels, and Equations and Tests.  The theory section describes the mechanical aspects and function of all the components of the wheel and gives the design philosophy and most of the common variations in wheel design.  The book discusses technical aspects of load and stiffness in layman’s terms without equations.  The purpose of the discussion is to get across the reason wheels are the way they and to get away from vague terms like “stiffness” and “responsiveness” as justifications for various wheel building design choices.  The theory section of the book gives technical explanations but unfortunately without delving into the mathematical aspects of stress and strain that are necessary for the truing algorithm.

Part 2, the wheel building section, gives a detailed procedure for assembling a wheel from the components: rim, hub, spokes, and nipples.  This section is the “how-to” part and is at the heart of message to the practical cyclist and wheel mechanic.  It covers the selection of components and tools, methods and designs for lacing the spokes, and the process of tensioning and truing the assembled wheel.  It describes how to determine the correct spoke tension.  The procedure is similar to information found elsewhere, but more complete and more technically sound, being based on engineering principles.  The choices are explained in terms of the concepts given in the theory section.

Part 3 of the book is a brief discussion of some experimental measurements on the effect of tying and soldering spokes and on the elasticity and elastic limits of swaged versus non-swaged spokes. The book includes a brief section on the equations for modeling spoke tension and provides some typical spoke and rim data and a sample calculation. The purpose of the spoke equations appears to be for designing wheels and selecting an appropriate tension in the spokes for the wheel building, not for truing or modeling. 

The final section describes, without giving any equations, a finite element model of a bicycle wheel. The tabular results of the finite element computer program’s inputs and outputs are given.  This form of presenting results harks back to a much earlier computer era in which computer results came out of line printers and were presented to the code’s author in thick stacks of green-lined, fan-fold paper.  I got a twinge of nostalgia just seeing it, but I would have rather Brandt used that space in the book to give his modeling equations.

The author apparently developed the finite element model to better understand the distribution of reaction forces in the spokes of the wheel loaded at the axle against the ground or loaded torsionally at the hub.  The results computed with the code are given in the main text to illustrate how the tension in spokes and bending of the rim are coupled together.

One of the discussions utilizing the finite element model in the theory section concerns the radial deflections due a load applied at the axle against the road.  The main point is to resolve the question of whether the load of the axle is more accurately described as supported from the point of contact or suspended from the spokes.  The analysis addresses how much load is required for the least stressed spoke to be less than zero (slack spoke).  The question was one of seemingly academic interest for the purposes of discussion with other wheel builders, some of whom argued that the axle “hangs” from the upper spokes or “stands” on the lower of the rim. The results, shown in Figure 1 from the book, shows that the deflection due to a vertical load on the axle is concentrated very closely around the contact patch (the point of the wheel in contact with the ground) and that very small deflections and loads were distributed around the rest of the wheel.  The figure supports that author’s claim that the wheel stands on its bottom spokes (more accurately, decreases their tension) rather than hangs from its upper spokes, since the deflection of the rim and change in tension of the spokes is much greater at the bottom than the top of the wheel. The fraction of the circumference that undergoes increased spoke tension is much larger but the magnitude of the change in tension very small and distributed rather evenly.

Figure 1.  Radial deflection of rim due to vertical loading calculated by Brandt

While the vertical loading problem is not the same as the truing problem, the deflection of the rim due to tensioning one single spoke more than others is roughly the same problem as an externally applied load.  (It is not exactly the same.  The external load of the road is distributed along the rim by the tire and contact patch; whereas, the spoke tension is concentrated at a point.) The results in Figure 1 show that the effect of adjusting a spoke would be localized around the spoke and the displacements of rest of the spokes would be much smaller.  Mathematically, the localized effect suggests that an algebraic linear model of the wheel will be nicely diagonalized and will therefore be amenable to stable and accurate mathematical computations.

Brandt’s book establishes him as one of the experts in wheel design and building but does not lead directly to any useful input for the development of a mathematical model of the wheel for our purposes (other than for the spoke model), but it does give insight into the technical issues in wheel design and building and a practical understanding of the methods of wheel building.

The late, great Sheldon Brown’s web site, www.sheldonbrown.com, has a technical entry on wheelbuilding, www.sheldonbrown.com/wheelbuilding.html.# The entry is useful to explain the wheelbuilding terminology, history, and methods of traditional wheel builders.  We used this site to better understand how to lace spokes correctly and to relate our method to the traditional method that relies on experience and feel. It has no technical information regarding the modeling of the wheel.

The Burgoyne and Dilmaghanian# article entitled “Bicycle Wheel as Prestressed Struction”, which we were able to access online, gives a brief history of bicycle wheel development and of modeling of the wheel.  Rather than developing a model, the main purpose of the article is to present a comparison of a mathematical models to Burgoyne and Dilmaghian’s experimental data.  The Pippard and Francis model from 1931# is chosen for comparison and the comparison is quite good suggesting this is an appropriate model to start with.  As in the Brandt article, the test case is an external load.  We have been unable to obtain a copy of the Pippard and Francis article (because we would have to pay too much for it) so it is not included in the background.  However, inferences from Burgoyne and Dilmaghanian and other articles (Hetényi) suggest that the Pippard and Francis model is similar to the circular beam utilizing the Euler-Bernoulli approximations for thin beams that we have chosen in this paper.  The Burgoyne article is a source of other references to spoked wheel literature including the Hetényi text which has proven instrumental in formulating the model needed for the problem and the Gavin article.

Gavin’s paper,# “Bicycle Wheel Spoke Patterns and Spoke Fatigue,” compares the stress in the spokes under a radial, tangential and axial loads for one-cross, two-cross, and three-cross spoke patterns.  He also addresses the relationship between stress and fatigue cycles, but most of the article is concerned with stress calculations and measurements rather than fatigue.  He references the Pippard model of a spoked wheel and the Hetényi text on beams on an elastic foundation.  He gives formulas which he derived based on Hetényi for the maximum deflection for a given a load.  He also states that he used a “three-dimensional elastic frame analysis” to evaluate the accuracy of his formulas.  I am not sure what a three-dimensional elastic frame analysis is but I believe it is a solution in which each segment of the rim between spokes and each individual spoke are treated as straight Euler-beams in transverse and axial loads.  It is implied that the curved beams are approximated as straight segments between the spoke nipples.  The equations and calculation method of the frame analysis are not described at the level of detail that would be useful for our modeling task.  However, he states that he uses his elastic frame analysis to evaluate the stresses due to spoke contact with other spokes, torsion, and tangential compression in the rim and finds all to be negligible compared to the effects of beam deflection in determining spoke stress. His conclusion about spoke fatigue, which is interesting but is not particularly relevant to our study, is that the spoke patterns are equivalent except for tangential loads (e.g. braking or pedal force) in which the greater number of crosses (greater tangential angle of the spoke with respect to the rim) reduces the maximum spoke stress.  One interesting fact is that Gavin states that he fitted experimental data for rim bending stiffness to his model rather than use a theoretical model of the centroid of the rim cross section.  This choice was dictated by the complexity of accounting for the effect of spoke holes in the rim. We expect to apply a fitting approach to estimate the bending constant of rims also for the same reason.  The details of his fitting routines are not described.

General references on Linear Elastics

Barber’s online document, Linear Elastostatics,# available at his personal web page at the University of Michigan, appears to be reference material for a class that Barber teaches as well as the starting point for his hardcover textbook, Elasticity, which we also used.  The introductory section on the stress and strain equations, compatibility relations and equilibrium equations, and the nomenclature are generally useful, particularly for use with the polar coordinates chapter of a Barber’s Elasticity book which is discussed next.

Linear Elastostatics covers three-dimensional stress and strain.  It provides notes on significant theorems that are applicable to stress and strain solutions and the Michell solution to a disk with a hole at the center.  It does not address the degenerate case of the slender beam and Euler-Bernoulli approximation that we ultimately used.  The Michell solution could be viewed as is a general representation of the wheel which models the strain and strain throughout the three dimensional structure than the one-dimensional slender beam approach for modeling the rim and spokes that we have chosen.

J.R.Barber also has a closely related, hardcover book, Elasticity#, which covers much of the same material as Linear Elastostatics.  Individual chapters of this book may also be purchased electronically online.  I only purchased Chapter 8, “Problems in Polar Coordinates.”  Chapter 8 covers the conversion of the general stress and strain equations from Cartesian to polar coordinates. The chapter discusses Michell’s solution to the elasticity problem in polar coordinates. Section 8.3 describes a solution method for a disk with a central hole using Fourier series expansion method.  The problem is closely related to the rim problem except that the radial dimension is finite rather than infinitesimal as in the Euler-Bernoulli beam approximation.  We were able to apply some of the same solution techniques in the rim problem.  For example, our solution uses the same approach to the tangential boundary conditions.  Barber’s statement about boundary conditions is that there is no tangential boundary and hence no tangential boundary condition, only the requirement that the solution must be single-valued and periodic.  The solutions for stress and strain must be equal at every 2

\(\phi ,\;\phi  + 2\pi ,\;\phi  + 4\pi ,\;\;...\)  The periodic requirement is also applicable to the slender circular beam model.  Another of Barber’s solution techniques that we followed was converting the forcing function in the problem into a Fourier series.  Barber transforms the radial boundary conditions into a Fourier series and then matches the coefficients of the particular solution of the differential equation to the coefficients of the transformed boundary condition using the method of undetermined coefficients.  This is similar to the method used in our solution.  The difference is that our forcing term is the set of spoke tensions acting on the rim which appears in the differential equation rather than in the boundary condition.  This difference is a result of reducing the rim to an infinitesimal radial dimension.

A number of Wikipedia articles were useful as an introduction to elastostatics and provided references to more detailed sources of information.

“Euler-Bernoulli Beam Theory,” article# gives the derivation of he Euler-Bernoulli beam equation for straight beams and gives a couple of solutions of elementary loadings and boundary conditions.   This article was the basis for the idea to use a one-dimensional rim but convert the geometry to a circular beam.

 “Michell Solution” article# gives a general solution of the two-dimensional linear elastics problem in \(\left( {r - \phi } \right)\) geometry.  When I found it, I thought the solution would lead toward a solution of the rim problem because it was an exact solution of the elasticity equations in polar geometry.  However, the article does not discuss how the two-dimensional solution reduces to a one-dimensional circular rim as desired for this application. The Wikipedia entry also includes Barber’s table of the stress components and a reference to Barber’s book which was the route by which we found the book.

Bauchau and Craig’s book# is a general text on structural analysis and has a significant chapter on Euler-Bernoulli beam equation with a number of useful examples.  It covers the conversion of the Euler-Bernoulli modeling approximations of a slender, straight beam to a slender beam that is curved in three-dimensions.   It is the most general statement of the Euler-Bernoulli problem that was found.  In the model derivation, the rim and spoke geometry is best stated in polar coordinates so the problem is formulated in those terms rather than Cartesian coordinates as in Bauchau and Craig.  This choice of coordinates leads to different but equivalent forms.

Hetényi’s text# is devoted to solutions and applications of a single problem, a slender beam with transverse loading like the Euler-Bernoulli beam equation with the added feature that the beam is in contact with an elastic foundation that supports the beam perpendicularly at every point. The basic equation was developed to model railroad track on an earth foundation.  Hetényi recognized that the same equation was applicable in many other physical problems.  Hetényi specifically discusses the spoked wheel and references the solutions of axial and radial loading problems solved by Pippard and Francis.  In the bicycle wheel problem, the wire spokes, approximated as a continuum, are the elastic foundation.  The spoke wire material is more nearly a linear elastic foundation than the earth is for the rail problem.  The difference in geometry between the straight railroad track and the circular rim is dealt with by recasting the original equation in polar coordinates.  My derivation of the circular beam derivation follows Hetényi’s except that I do not approximate the spoke tension as a continuous force density function.  This actually results in a different form of differential equation and requires a different method of solution.  In Hetényi’s formulation, the rim and spoke models are combined in the differential equation.  In my solution the rim and spokes are model is a separate systems of equations which solved separately and then combined algebraically as systems of linear equations.  The approach I have taken represents the spokes as individual point loads so that each input to the problem appears explicitly without an unnecessary approximation.   Another difference is that only the radial loading problem is presented in Hetényi. He alludes to the existence of an axial loading model by Pippard and Francis but it is not shown.   I derived an axial loading model by following the same approach as the radial problem. As it turns out, the axial equation is the same as the equation for a straight beam if the shear terms in the rim stiffness matrix are neglected.

Lots of Equations to Upload

I am about finished with the model. I still think it is just too much work to upload equations into blogger.  I will try to either figure out an easier way or just slug it out.

Introduction to the Bicycle Wheel Truing Algorithm Problem

Bicycle Wheel Truing Algorithm

Introduction

A bicycle wheel is a pre-stressed system consisting of rim, spokes, and hub. The tensioning of the spokes pre-stresses the system so that the spokes remain in elastic tension (not slack) under the loads of bicycling.  Tensioning also accounts for any abnormalities in the circular shape of the rim. Varying the tension in individual spokes during the truing process pulls the rim into radial and axial trueness and corrects for any out-of-roundness in the untensioned rim. The process is accomplished using a wheel truing stand on which the wheel can be rotated past a fixed point that measures how much the rim is out of round (the deflection) in both the axial direction (in the direction of the axle, or in other words, lateral direction between the brake pads) and radially.  The particular truing stand used in this study is a Parks TS-2.2 Professional Truing Stand using Dial Indicators TS-2DI to provide measurements of the deflections in axial and radial directions.

Figure 1:  Parks TS 2.2 Professional Truing Stand

Figure 2:  Close-up of TS-2DI Dial Indicators

The wheel building system also includes a spoke tension meter and a wheel dish measurement tool.

Figure 3:  Wheel Dish Tool and Tension Meter

The tension meter is a device that measures the tension in the spoke by pressing a calibrated spring against the spoke.  The deflection of the spring is proportional to the tension in the spoke.  Figure 4 and Figure 5 illustrate the operation of the tension meter.

Figure 4:  Tension Meter applying pressure at three points flexing the spoke

Figure 5:  Tension meter scale showing reading from measurement

The dish-measuring tool is a C-shaped beam that has an adjustable point of contact at the center.  The tool is pressed alternately on both sides of the rim to see that the axial distance from the rim to tip of hub axle is the same on both sides of the wheel.  This tool is used to ensure that the rim is precisely at the center of the hub axle.

The other objective in wheelbuilding, besides trueness of the rim, is to tension the spokes as nearly equally as possible, and with sufficiently high tension that the spokes do not become slack under vertical load or braking.  The tensioning pre-stresses the wheel in much the same way that steel reinforced concrete is pre-stressed so that the concrete remains in compression when loads are applied.  (The wheel is the opposite actually.  The pre-stress in the wheel ensures that the spokes remain in tension whereas the pre-stress in concrete ensure that it remains in compression.) Spokes that are cycled between slack and tensioned state are subject to metal fatigue and early failure. 

This particular analysis is concerned with developing an algorithm to aid in wheel building and maintenance to bring a wheel from the as-found condition to a true, round shape with proper tension in the spokes.  The objective is to use model-based control theory to design a spoke adjustment algorithm.  The measured axial (lateral) and radial deflections and the spoke tensions are entered into the truing algorithm and the spoke nipples are adjusted by amounts determined by the control algorithm to bring the rim into a true, equally tensioned, final state.  The actual measuring of the deflections of the rim and the tension in the spokes, entering them into a spreadsheet, and adjusting the spoke nipples are manual.  The automated portion is the computation of the number of turns.  The process is that the wheel mechanic reads the dial indicators and spoke tension meter at each spoke position and inputs the data manually into a spreadsheet.  The spreadsheet then computes the number of turns of the spoke nipples for the next iteration of the truing process.  The wheel builder then adjusts each spoke individually by the amount computed by the algorithm to reach the desired setpoints for trueness and tension.  The process is repeated until the rim is within the desired tolerances of trueness and tension.

The first step in the problem investigation is the wheel model.  The wheel model defines the mathematical relationships between are the deflections of the rim and tension in the spokes and the position of the spoke nipple using linear elastostatic models of the rim and spokes.  The variable inputs to the model are the rotational positions of the spoke nipples.   The fixed data are the elastic properties of the rim and spoke materials and the wheel geometry.  The outputs are the tensions in each spoke and the radial and axial deflections from round shape.

In the background research, we seek suitable mathematical models of the elastic behavior of the rim and spokes for the truing problem if they exist or, if not, to find related models of the rim and the underlying theory of structural analysis that will allow us to develop a model that does suit the needs of the truing problem.  In our search, we have not found a model that closely fits the needs of our application; however, we were able to find sufficient references to develop our own model. 

The truing algorithm is developed from the field of model-based control. The control algorithm uses the mathematical model of the rim and spokes described above to compute a new setting for the spoke nipples given the current deflections of the rim and tension in the spokes.   The control algorithm is, in effect, a method of inverting the model, such that given the deflections and tensions, it computes the number of turns of the nipples to bring the wheel into true and equally tensioned spokes.  The mathematical difficulty of the control problem is that, as stated, the truing problem is mathematically over-determined.  That is, there are more setpoints  (axial and radial deflections and tensions to control to setpoints) than inputs (spoke nipples to turn).   Moreover, the goal of a perfectly round wheel and equally tensioned spokes is not possible for a rim that is not perfectly round in its unstressed condition.  The rim requires unequal tension to bring the shape to round. A strict mathematical inversion of the problem that computes the turns as a function of the deflections and tensions will not work.  Additionally our method must account for errors.  Neither our model nor our measured data are perfect.  Hence, the truing algorithm must also be robust to modeling and measurement errors.  The over-determined mathematical problem and the robustness to errors indicate that the truing algorithm should minimize a weighted sum of the squares of the deviations of the measurements from their set points (the least squares method) and follow a gradual to approach toward the setpoints in a number of partial steps (relaxation method).  Each step reduces the deflections and increases the spoke tensions a fraction of the measured difference between the current measured value and the desired value at each step. The least squares method and the relaxation method both add stability to the convergence and robustness to modeling errors.  The least squares method gives a way of dealing with the over-determined problem.

The weighted sum of squares is usually called a quadratic cost function.  The quadratic form is convenient because the minimization of the quadratic results in a mathematically elegant, linear problem which can be readily solved.