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.
