Background on Bicycle Wheel Modeling and Truing Controls 

No prior work in modeling a spoked wheel has reported an analytical solution of the wheel model by the approach taken here.  A number of authors have addressed  modeling of the spoked wheel but not a general three-dimensional, analytical model in which individual spokes are represented as inputs.  Several authors have modeled the bicycle wheel for the purposes of examining the effect of an external load rather than effects due to adjusting the spoke nipple inputs.  These related models fall into two groups, the analytical models (Hetényiv and Pippardvi) and finite element (Brandti, Ngvii, Hartzviii, and others).  The deformation equations which are reported in these models have two serious deficiencies from the standpoint of the actual physics of the model in two respects.  First, the hoop stress due to the net effect of spoke tension on the rim is not considered in the deformation equations of the rim.   Second, the coupled effects of twist and lateral deformation are not considered.  The modes of twisting and lateral bending in a curved beam are different from straight beams.  In straight beams, the twisting and bending modes of deformation are not coupled.  In curved beams they are coupled.  Hence, even when no external twisting torque is applied to the curved beam, lateral forces cause both lateral bending and twisting. The analytical wheel models also represent the spokes as a smeared out sheet of elasticity rather than as individual point loads.  This form of the spoke model precludes the individual adjustment of spokes needed for the truing application.  The finite element models which have been reported are only two-dimensional models.  Only inplane deformation to radial load is calculated.   These models would not meet the need of the truing application.  Certainly, finite element methods could be used to model three dimensions and twist, but these examples in the literature do not have that level of generality.  The general purpose codes for structural analysis such as ANSYS lack the torsional mode of deformation.

The matrix form of the solution is also different from previous wheel models.  The matrix form is appropriate and convenient for control analysis and simplifies both the reporting and the programming of the solution.  In effect, the programming is exactly the same as the derivation.

The following sections describe the prior modeling work and the shortcomings for use in the control algorithm.  

The bicycle models found in the literature all fall short of representing the physical effects that are present in the deformations associated with truing and tensioning a bicycle wheel.  The initial modeling effort in this research followed the analytical approach of Hetényi and Pippard.  Comparisons of measured to predicted cases did not match very well.  The predicted wheel was stiffer than the actual measured wheel.  Additional literature searches for more fundamental models located a group of papers that discuss buckling of curved beams and the concept of flexural-torsional buckling.  These papers do not include any applications of the equations to the bicycle wheel.  The background on these papers is covered in a separate section.  This flexural torsional approach proved successful in modeling the observed behavior and is the basis of the current wheel model.

One prior report of a control algorithm for spoked wheel truing has been found.  This method uses an experimental determination of the influence function.  This algorithm was patented in 1993 and is also reported in the background following.

Bicycle Wheel Modeling References

The book by Jobst Brandt, The Bicycle Wheel,^[1] 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 reasons why wheels are built the way they are and to get away from vague terms like “responsiveness” and technically inaccurate usage of “stiffness”. The theory section of the book gives technical explanations but it does so unfortunately without delving into the mathematical aspects of stress and strain that are necessary for the model-based truing algorithm.

The final section describes, without giving any equations, a finite-element, mathematical model of a bicycle wheel.  The model is two-dimensional in the plane of the rim.  The model does not simultaneously solve the axial, radial, and twist displacement due to spoke tension.  The model is intended for calculating the effects of external load rather than the effect of individual spoke nipple adjustments.  The tabular results of the finite element computer program’s inputs and outputs are given.  No comparisons to experimental results are given.

The Burgoyne and Dilmaghanian article entitled “Bicycle Wheel as Prestressed Structure” gives a brief history of bicycle wheel development and of modeling of the wheel.^[2]   Rather than developing a new model, the main purpose of the article is to present a comparison of a mathematical model to Burgoyne and Dilmaghanian’s experimental data.  The Pippard and Francis model from 1931^[3] is chosen for comparison which turns out to be quite good.  As in the Brandt finite element model, the problem being considered is an external load rather than adjustment of individual spokes.  The inplane response to a radial load is the deformation least affected by the deficiencies of the Pippard and Francis modeling approach listed above.

Gavin’s paper,^[4] “Bicycle Wheel Spoke Patterns and Spoke Fatigue,” compares the stress in the spokes under radial, azimuthal, and axial loads for radial, 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 formulae which he derived based on Hetényi’s model for the maximum displacement for a given a load.  He also states that he used a “three-dimensional elastic frame analysis” to evaluate the accuracy of his formulas.  He reports fitting the beam stiffness coefficient of the rim to experimental data for the rims under test.  He explains this approach dictated by the complexity of accounting for the effect of spoke holes in the rim.  It also could be that he found that Hetényi’s model overpredicted the stiffness as we found in this report.  By fitting the coefficient, he improved the comparison but obscured the deficiency in the model.

Hetényi’s text^[5], Beams on Elastic Foundation, 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 a simplification to the analytical wheel problem apparently suggested by Southwell, Pippard and Francis approximated the wire spokes as a continuum with a single, fixed elastic constant around the rim representing the spokes.  This approximation, while allowing a closed form analytical solution to the problem in one dimension, loses the capability needed for the truing problem which requires representing the tension in individual spokes. .

In the late 1920’s and 1930’s, A. J. S. Pippard collaborated with J. F. Baker, M. J. White, and W. E. Francis on analytical structural models of circular rings and arches with and without spokes and bracing to represent frames of dirigibles, spoked artillery wheels with rigid spokes, wire spoked wheels, girders, masonry arches, and outer similar geometries. The modeling was reduced by approximations to forms suitable for hand calculation.  The research included experimental confirmation of many of the equations. The original research was reported in a series of papers in Philosophical Magazine and various government research reports.  In 1952, Pippard collected the papers into a monograph, Studies in Elastic Structures,^[6] which is the primary reference by which we have obtained Pippard’s work.  Two approaches are reported for modeling spoked wheels, the continuum approximation in which the set of spokes is approximated as an elastic sheet and an exact solution with individuals spokes.  The continuum spoke approximation solves for single radial and tangential point loads, but not axial loads.  General problems with multiple external loads could be modeled by superposition.  The continuum problem however does not lend itself to the complexity of tuning the wheel using individual spokes.

The exact method depends on an integral of a segment of the rim containing a single spoke.   The segments are coupled by continuity conditions at the ends of the segments.  The combined system is solved as a system of coupled equation.  The exact model with individual spokes is only solved for low numbers of radial spokes (3, 4, and 6 spokes) and would be cumbersome and inaccurate to extend to the number of spokes in typical bicycle wheel geometries.  Pippard’s quote explains the method’s limitations.

…the equations derived … provide a complete solution to the problem, but the work involved is laborious and very great accuracy is necessary since the final equations yield large numbers which differ by only small amounts.

Modern computers could deal with the algebraic difficulties of solving the multi-spoke problem by this method but would still be limited by the numerical stability of the formulation.  The method would appear to be a generalized finite element method in which the “element” is a segment of rim and a spoke that is solved analytically and coupled through end conditions to segments on either side.  While the equations have not been programmed and tested by the author of this report, this numerical structure is inherently subject to larger round-off error than the approach in this paper involving a trigonometric series solution.

Finite element models of the spoked wheel were produced by Ng as her master’s thesis^[7] and by Hartz as a class project^[8]. Ng’s paper considers the stiffness of the wheel as a function of the number of spokes while Hartz’s is an exercise in the use of ANSYS software.  Hartz develops an ANSYS model and compares his results to the experimental and computational results reported by Burgoyne and Dilmaghanian. Hartz reports some deficiencies in the comparions which he was unable to resolve in the context of the assignment.  Based on our own experience, the deficiency would likely be related to neglecting the torsional component of the deformation.

Ford reported a finite element solution of the bicycle wheel in an online record of projects he conceived as a graduate student at Northwestern^[9].  The details of the modeling are only available as coding.  Coding is, in general, a difficult format to read and assess so this model has not been reviewed in a critical way.

Bending of curved beams

In researching the literature for beam bending, we find that the theoretical modeling of curved elastic beams is a well-developed area with a number of journal articles and textbooks addressing this particular problem.  The models are directed primarily at calculating the critical or buckling load.  In the early part of the century, it was discovered that curved beams buckled at much lower levels than predicted by the Euler-Bernoulli beam calculations and studies into the deficiencies in the beam model were undertaken.  In 1961, Timoshenko and Gere described the problem of buckling for curved beams and rings and gave a complete history of the development up to that point.  Timoshenko and Gere noted that for straight beams if the lateral load is applied in a direction that passes through the neutral axis then the torsion is zero and the rotation is zero.  However, they recognized that this result does not hold for curved beams with azimuthal compression.  Nor does it hold for nominally straight beams with slight initial curvature.   Following the approach of Euler in which a small deformation is assumed, they applied equilibrium conditions for force and moment to obtain a system of three differential equations involving the radial, axial, and torsional displacements.  When torsion and flexure of the curved beam are accounted for in the structural model, buckling occurs at a much lower load than would be predicted by flexure alone.  Despite getting the main effect of coupling between flexure and torsion correct, Timoshenko and Gere’s analysis failed to meet some logical tests of consistency with the Euler-Bernoulli assumption that the strain in the profile of the beam is zero.  Further refinements were added to correct the deficiency.

Vlasov also published a paper in 1961 regarding elastic instability in curved beams that is widely cited as one of the seminal works in the field.  The original report has very limited availability and is not available at all online.  This author was unable to obtain a copy and defers to other authors listed in the references who cite his work for their discussion.

In the 1990’s and early 2000’s, the structural mechanics of curved beams continued to be refined, adding terms to the formulation of longitudinal strain across the profile of the beam as a function of the deformation of the beam at the centroid that made the deformation more consistent and presumably more accurate.  The works of [Yang and Kuo 1987], [Trahair and Papangelis 1987], and [Rajasekaran and Padmanabhan 1989], and [Bradford and Pi 2006] have converged on a technique for solving the problem and their works have each developed a viable, though not necessarily identical, solutions.  A consensus has emerged on the features of the equations of curved beams.  In this author’s view, the method of derivation was largely unified under the general solution of Bradford and Pi.  Their solution applies the minimum virtual work approach rather than the expressions of force and moment equilibrium for a differential element of the beam as Timoshenko and Gere applied.  Bradford and Pi present for comparison a derivation using the equilibrium equations for force and moment to show that the same solution is obtained either way.  They only presented equations for the axial and twisting deformation because they were interested the lowest level buckling which involves only those modes of deformation.  We have supplemented the system model with the equation for radial deformation from Timoshenko and Gere.

The derivations of [Yang and Kuo 1987] and [Rajasekaran and Padmanabhan 1989] are useful because they carry the derivation for classical set of deformation equations in four dimensions that gives the differential equations for beam deformation for radial, azimuthal, axial, and torsional directions (u, v, w, ) whereas Bradford and Pi only give the equations for axial bending and torsion.  The utility of these two papers is somewhat compromised by the number of typographical errors and the lack of complete statement of approximations used.

Wheel truing algorithms

One significant prior work in the area of truing algorithms is known, Papadopoulos’ patent^[10] of a wheel truing control algorithm.  The intent of his patent is the same as this paper, and the least squares error reduction is similar to the approach in this paper.  The main difference is that, in his solution, Papadopoulos uses experimentally determined influence functions rather than an analytical model of the rim and spokes.  The experimental influence function is the difference between a reference state and a perturbed state in which one spoke is tightened by a known amount.  The experiment is difficult to do with adequate precision because the resolutions of the displacement and tension measurements are on the same order as the differences caused by the perturbation.  The measured data also do not conserve quantities such as mechanical equilibria which are conserved in the analytical model.  Hence, the analytical influence function is generally superior to the experimental.

Papadopoulos’ method computes the control gain matrix from the experimental influence function using a least squares error minimization that is similar to the method in this paper.  The difference, and it is significant, is that Papadopoulos minimizes the sum of the squared, predicted control error whereas this paper takes the more common control approach in which the performance index includes the squares of the predicted control errors and the squares of the predicted spoke adjustments. Including the spoke adjustments in the performance index gives a means for controlling the rate of approach to equilibrium and gives the control algorithm an adjustable degree of robustness to measurement and modeling error.  The approach used in this report is the generally accepted method for modern, multivariate control and is called the linear quadratic regulator. (refs ???)