## 3 Results and Discussion
### Various Sizes of Mountain Bike Tires
#### 3.1.3 Forces and Moments  
Force and moment plots based on the Pacejka Magic coefficients and FastBike fit parameters are shown in Figure 6. Although each dataset retained a specific marker type, its line style was divided into two sections. The solid line portion, emanating from the origin, indicates values within the measurement range, whereas the dashed portion of the line represents any extrapolated values on the basis of the fit. As described previously, the limited treadmill width of the test fixture constrained slip and camber ranges, particularly for the wider knobby tires. Although the linear portion of the curve near the origin, whose slope (i.e., stiffness) was represented by the product of BCD Pacejka coefficients, was well identified, the curvature toward the peak and the eventual asymptote often had lower resolution, as was evident when comparing the \(29\times 2.3\)\" solid versus dashed line to that of the 26x4\". As such, caution should be exercised if attempting to examine peak or curvature behavior of the wider tire variants, especially in terms of slip angle. Figure 6: Force and moment fitted curves for \(29\times 2.3\)”, \(27.5\times 2.8\)”, \(29\times 3\)”, and \(26\times 4\)” knobby tires with (**a**) normalized lateral force vs. slip angle, (**b**) normalized lateral force vs. camber angle, (**c**) normalized self-aligning moment vs. slip angle and (**d**) normalized twisting torque vs. camber angle. Upon examining each of the plots in Figure 6, several interesting trends were evident. In Figure 6a, the upper left plot of normalized lateral force versus sideslip, the \(29\times 2.3\)\" had the lowest cornering stiffness. The \(27.5\times 2.8\)\"and \(29\times 3\)\" plus tires shared similar, steeper slopes but differed in curvature, whereas the \(26\times 4\)\" fat bike tire had nearly double the cornering stiffness of the baseline tire. The vertical axis was limited at a normalized force of 1.0 to highlight another interesting trend. Although the peak of the curve, whose magnitude was captured by the Pacejka D coefficient, should represent the peak frictional value (approaching 1.3 here), the tires were tested on a non-skid tape (a particular formulation of sandpaper)-coated treadmill, not on actual asphalt or dirt, which would have a commensurately lower coefficient of friction. Hence, as with most test bench characterization data, it is useful for relative comparison, while various scaling methods can be employed to adjust to an appropriate friction level if desired. In addition to the four knobby tire curves, whose camber stiffnesses also increased with width (Figure 6b), the upper right plot of normalized lateral force due to camber also included a reference line representing the so-called tangent rule, at which point the net ground reaction force, without slip, is in the plane of the wheel [7], as shown by  
\[\frac{F_{y}}{N}=\tan(\varphi), \tag{6}\]  
where \(F_{y}\) is the lateral force, \(N\) the normal load on the tire, and \(\varphi\) the camber angle. Any tire below this line, such as the \(29\times 2.3\)\" knobby, must make up the difference via a positive sideslip angle (into the turn). The other three tires would need negative sideslip angles (slipping to the outside of the turn) for equilibrium. Depending on the tire pairing, front versus rear, this may influence the vehicle's understeer/oversteer ratio. The lower left plot (Figure 6c) shows the normalized self-aligning moment, which acts to align the wheel with its velocity vector, that is, diminish the tire's slip angle. In various physical tire models, this torque can be thought of as the product of the lateral force and pneumatic trail, where the pneumatic trail is the result of an offset in the shear stress distribution toward the rear of the contact patch. An approximation of the tire pneumatic trail is shown in Figure 7, obtained by dividing the aligning moment by the lateral force due to slip, and ignoring the point at zero slip for which no lateral force exists.