## 3 Results and Discussion
### Various Sizes of Mountain Bike Tires
#### 3.1.2 Footprints  
As described in the various literature on cars [8] and motorcycles [9], tire footprints can tell us a lot about the interaction between the tire and ground. Normal load is applied to the tire, which deforms to create a contact patch, or footprint, which at zero camber angle is quite elliptical on smooth,  
Figure 4: Cross sectional profiles for \(29\times 2.3\)”, \(27.5\times 2.8\)”, \(29\times 3\)”, and \(26\times 4\)” knobby tires, respectively. flat ground. The distribution of the normal load over the contact patch area yields the contact patch pressure (not measured here). In the absence of camber, the contact patch pressure may take on a fairly regular distribution. The presence of tread knobs obviously localizes contact pressure. Similarly, the knobs localize the shear stress used to generate lateral and longitudinal friction forces as the tire is slipped, cambered, driven or braked. Although not the focus of this work, it has been suggested that even for car tires, depending on (more closely spaced) tread pattern, the localized pressure can be \(10\times\) the average [10]. Figure 5 presents the knobby mountain bike tire footprints in the same order and on the same 10 mm grid as the cross sections in Figure 4. Here, the fitted ellipse was superimposed on the footprint and its axes were used to calculate length and width of the contact patch. Examining overall dimensions, it can be seen that the \(26\times 4^{\prime\prime}\) fat bike tire had both the longest and widest contact patch followed in length by the \(29\times 3^{\prime\prime}\) plus tire, the \(29\times 2.3^{\prime\prime}\) knobby, and \(27.5\times 2.8^{\prime\prime}\). Hence, the length of the contact patch depended not only on tire outer diameter, but presumably also on deflection, as influenced by inflation pressure, vertical load, and the tire's carcass. In terms of contact patch width, the \(26\times 4^{\prime\prime}\) was widest, whereas \(27.5\times 2.8^{\prime\prime}\) and \(29\times 3^{\prime\prime}\) were essentially equal, despite their size designation, and the \(29\times 2.3^{\prime\prime}\) at its nominal inflation pressure was narrower. These images bring very clear meaning to the tire industry term \"void ratio\" [11]. For these knobby tires, the amount of white, empty space within the contact patch ellipse considerably exceeded the amount of black tread interacting with ground. The same MATLAB script used for fitting the ellipse also computed the void ratio by comparing white space to contact patch area after converting the image to binary. The resulting void ratios in terms of percentage for each knobby tire are listed on the figure and ranged from 69.7% to 80.6%. As noted in the cross-sectional profiles, the tread pattern of the \(27.5\times 2.8^{\prime\prime}\) and \(29\times 3^{\prime\prime}\) tires essentially scaled the size and spacing of the \(29\times 2.3^{\prime\prime}\) tread pattern upward in proportion to tire size. Again, the tread-pattern of these three tires offered no center ridge of knobs, whereas the \(26\times 4^{\prime\prime}\) tire did. Note that the three tread patterns on the left employed essentially two different knob typologies, one that was rectangular and one that was slightly wedge-shaped with an hourglass-shaped longitudinal groove whose depth was roughly half of the 4.5 mm tread depth. The \(26\times 4^{\prime\prime}\) fat bike tire, on the right, showed four knob typologies within its footprint, two alternating central knob chevrons with varying degrees of central and trailing-edge relief, and two types of intermediate knobs similar to those of the other tires.