Loading of Bicycle Wheels [14]

The bicycle wheel is subjected to high static (weight of the rider) and dynamic (inertia on road bumps, torques for acceleration, braking, and traction, etc.)

forces. Its predecessor—a wagon wheel (Fig. 3.22)—has relatively strong wooden spokes and rim. The spokes in the lower part of the wheel accommodate (by compression) the loads transmitted to them by the rim and the hub, just like the bearings discussed earlier. The bike’s wheel needed to be much lighter, thus wood was replaced by high strength metals that allowed one to dramatically re-duce cross sections of both the spokes and the rim. In fact, the spokes possess necessary strength while having very small cross sections equivalent to thin wire (Fig. 3.23). However, such thin spokes cannot withstand compression due to buckling. The solution was found in prestressing the spokes in tension, so that the tensile preload force on each spoke is higher than the highest compressive force to be applied to the spoke during ride conditions. With such a design, the spoke would never loose its bending stiffness if the specified loads are not ex-ceeded. However, the lateral stiffness of the spoke is decreasing when the tension is reduced by high radial compressive forces (see Section 7.4.1). In such a condi-tion, a relatively small lateral force caused by turning, for example, may lead to collapse of the wheel.

Contrary to the antifriction bearings which were discussed earlier, the rim is a relatively compliant member. Since the total force applied to the rim by the prestressed spokes can be as great as 5000 N (1100 lb), the rim is noticeably compressing, thus reducing the effective preload forces on the spokes. Pressur-ized tires also apply compressive pressures to the rim equivalent to as much as

Figure 3.22 Spoked wagon wheel.

Figure 3.23 Spoked bicycle wheel: (a) front view; (b) side view.

7–15% of the spoke load. The tension of the spokes changes due to driving and braking torques, which cause significant pulling and pushing forces on the oppositely located spokes. Combination of the vertical load with torque-induced pushing and pulling loads results in local changes in spoke tension, which appear as waves on the rim circumference. These effects are amplified by the spoke design as shown in Figs. 3.23: to enhance the torsional stiffness, the spokes are installed not radially but somewhat tangentially to the hub (Fig. 3.23a); to en-hance lateral stiffness and stability of the wheel, the spokes are installed in a frustoconical manner, not in a single plane (Fig. 3.23b).

When a vertical load is applied to the wheel, the spokes in the lower part of the wheel are compressed (i.e., their tension is reduced).The spokes in the upper half of the wheel are additionally stretched, but even the spokes in the midsection of the wheel are increasing their tension since the rim becomes some-what oval. This ovality is not very significant—the increase of the horizontal diameter is about 4% of the deformation at the contact with the road—but it has to be considered. Since the rim is not rigid, its flattening at the contact with the road leads to reduction of effective stiffness of the wheel. It is in agreement with

a suggestion in Section 3.5.1 to enhance stiffness of a bearing by increasing the vertical diameter of the bore.

It is important to note that performance loads (radial, torque, braking, and turning loads) cause significant distortion of the rim (Fig. 3.24), which, in turn, results in very substantial deviation from load distribution for an idealized model (like the bearing model inFig. 3.21).

While the problem of load distribution in and deformations of the real-life spoked bicycle wheel is extremely complex, there is a closed-form analytical solution for load and bending moment distribution along the wheel rim and for deformations of the rim with some simplifying assumptions [15]. These assump-tions are as follows: the road surface is flat and rigid; the spokes are radial and are coplanar with the rim; and the number of spokes n is so large that the spokes can be considered as a continuous uniform disc of the equivalent radial stiffness.

With these assumptions, the problem becomes a problem of a radially loaded ring with elastic internal disc. Every point of the rim would experience radial

Figure 3.24 Distortion of bicycle wheel rim under radial and torque loads.

reaction force from the spokes proportional to deflection w at this point. There are n/2πR spokes per unit length of the rim circumference (R⫽ radius of the wheel). Deformation w and bending moment M along the rim circumference due to vertical force P acting on the wheel from the road are

w⫽ PR³

4αβE1I冢^{2παβa2 ⫺ A coshαφcosβφ⫹ B sinhαφsinβφ}冣 ^(3.30)

M⫽ ⫺PR

2 冢^{π1a2⫹ A sinhαφsinβφ⫹ B coshαφcosβφ}冣 ^(3.31)

The force acting on a spoke is obviously

Ps⫽E2F

R w (3.32)

Here E1and I⫽ Young’s modulus and cross-sectional moment of inertia of the rim, respectively; E2, F, and l ⬇ R ⫽ Young’s modulus, cross-sectional area, and length of spoke; a ⫽ (R²n/2π)(E2F/E1I ); α ⫽ √(a⫺ 1)/2; β ⫽

√(a⫹ 1)/2; and angular coordinateφof a point on the rim is counted from the top point of the wheel. Formulas (3.30)–(3.32) allow one to evaluate, at least in a first approximation, the influence of various geometric and material parameters on force and bending moment distributions. Figure 3.25 [15] shows these

distri-Figure 3.25 Calculated distribution of (a) bending moment and (b) radial forces along the circumference of a bicycle wheel.

butions for R⫽ 310 mm; I ⫽ 3000 mm⁴; n⫽ 36; spoke diameter d ⫽ 2 mm (F⫽ 3.14 mm); E1⫽ E2 ⫽ 2 ⫻ 10⁵N/mm²(steel). After the maximum static and dynamic forces on the wheel are estimated/measured, the data from Fig. 3.25bcan be used to specify the necessary initial tension of each spoke Pt. This tension should be safely greater than the maximum possible compressive force acting on the spoke, which for the parameter listed above is 0.280 Pmax.

3.5.3 Torsional Systems with Multiple Load-Carrying