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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.
