Reinventing the Wheel: Stress Analysis, Stability, and Optimization of the Bicycle Wheel

Ford, Matthew

doi:https://doi.org/10.21985/n2-3sh0-m798

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Reinventing the Wheel: Stress Analysis, Stability, and Optimization of the Bicycle Wheel

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The tension-spoke bicycle wheel owes its stiffness and strength to a cooperative relationship between the rim and the spokes: the rim holds the spokes in tension to prevent them from buckling under external loads, while the spokes channel external forces to the hub and prevent the rim from becoming severely distorted. The prestressed design enables the slender spokes to support compressive loads without going slack, but also makes the rim susceptible to buckling under compression. I aim to uncover the principles governing the deformation and stability of the tension-spoke wheel subject to internal and external forces. I establish a theoretical framework in which the wheel is modeled as a monosymmetric elastic beam (the rim) anchored by uniaxial elastic truss elements (the spokes) to a rigid foundation (the hub). From a general statement of the total energy of the system, I derive a set of coupled, linear, ordinary differential equations describing the deformation of the wheel and illustrate instances in which those equations can be solved analytically. To solve the general equations, I approximate the displacement field with a finite set of periodic functions to transform the differential equations to a linear matrix equation. This matrix equation leads to an intuitive model for calculating the lateral stiffness of the bicycle wheel by constructing an infinite array of springs connected in series, where each spring is associated with a discrete deformation mode. The series-springs model reveals the importance of the rim torsional stiffness, which is generally much smaller than the bending stiffness and therefore dominates the overall flexibility. The theoretical framework incorporates the effects of spoke tension, which can both promote wheel stability by preventing spokes from going slack, and reduce wheel stiffness due to the resulting compression in the rim. Contrary to both popular belief and expert consensus, increasing spoke tension reduces the lateral stiffness of the wheel, which I demonstrate through theoretical calculations, finite-element simulations, and experiments. I derive an equation for the maximum tension that a wheel can support before buckling. Two well-known buckling solutions emerge as special cases of the general wheel buckling criterion. Under external loads, two competing failure modes govern the elastic stability of the wheel: spoke buckling and rim buckling. The trade-off between spoke stability and rim stiffness leads to an optimum spoke tension of roughly 50% of the critical buckling tension in order to maximize the lateral load a wheel can withstand before spokes go slack. Using a machine designed and built by Northwestern undergraduate students, we test the strength of wheels under radial compression. By considering separately the two failure modes of spoke buckling and rim buckling, I develop a simple formula to predict the radial strength that matches our experimental result to within 10%. Finally I discuss the existence of optimal wheel configurations and properties. By reducing the design space to a single parameter---the mass of the rim divided by the total mass---I find optimal wheels which maximize the lateral stiffness, radial strength, or buckling tension. In general, more mass should be invested in the spokes when optimizing solely for lateral stiffness, while the rim and spoke mass should be on the same order when optimizing for strength and maximum tension. The existence of an optimal wheel for a given mass, rim radius, and hub width permits investigation of general scaling laws governing stiffness and strength. The strength of the wheel with respect to buckling under radial loads is proportional to the mass divided by the radius. Therefore the strength-to-weight ratio of the wheel scales with 1/R. Smaller wheels are inherently stronger relative to their weight than large wheels.

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