## 1 Introduction  
Mountain biking is a popular recreation and fitness activity that uses a bicycle and components designed to be rugged; to withstand off-road riding; and capable of handling unpaved surfaces, loose dirt, gravel, mud, and other terrains. In 2018, some 8.69 million people participated in mountain biking in the U.S. alone [1], which saw some USD 577.5 million in mountain bike sales the year prior [2].  
Tire behavior is a critical factor in bicycle performance and safety. Like road or city bicycle tires, weight must be kept low because, except for e-bikes, the rider must propel the vehicle under their own power. Tire durability is important because a flat tire can ruin a ride. Ride comfort, gleaned from the tire deflection, is a consideration even for bicycles with front and rear suspension, whereas performance and grip become even more important when navigating up or down steep grades, dodging trees, and other obstacles. In addition to tire size options, a wide variety of tread patterns, made of up individual \"knobs\", that is, tread elements, of various shapes, sizes, and depths are available, depending on intended usage.  
As with any sport, mountain biking has a large cadre of enthusiasts. There is much debate among racers, riders, and industry marketing lobbies over optimal tire size, tread pattern, inflation pressure, and, more recently, rim width. Although all these things are likely to affect a tire's performance, little or no scientific study of mountain bike tire properties exists in the literature.  
In this work, four modern mountain bike tire sizes were characterized through force and moment measurement via the tire test device at the University of Wisconsin-Milwaukee [3]. [PERSON]'s Motorcycle Magic Formula tire model [4], which emphasizes the high camber achieved by single-track vehicles, was fitted to the data while a non-linear polynomial from Dynamotion's FastBike motorcycle tire model [5] was fitted to the twisting torque due to camber. The four sizes of knobby tires were tested at realistic inflation pressures for their respective size and intended use on modern (wider) rim widths. Tire cross-sectional profiles were captured and compared using a simple and effective method. Tire footprint analysis comparing contact patch area as well as shape via the major to minor axes of the fitted ellipse was carried out. Static lateral and radial stiffness were also measured. Further comparison was made between the full knobby tire and a less-treated tire of similar carcass dimensions. These same tires then had their respective knobby and file-tread patterns removed to further quantify the tread influence. Trends in key tire properties across a range of inflation pressures were presented, along with results for two of the tires of interest on a slightly narrower rim.  
Mountain bike tires are commonly marketed in Imperial units, with the first number indicating the approximate outer diameter in inches and the second number indicating the approximate overall tire width when mounted. Although the metric-based, ISO 5775 international sizing designation developed by the European Tyre and Rim Technical Organisation (ETRTO) is generally listed somewhere on the tire's sidewall, its prominence seems to vary depending on brand. Because all tires in this study prominently displayed sizing in Imperial units and maximum recommended inflation pressure in pounds per square inch (psi) on their sidewalls, this work refers to the tires as such. Table 1 shows both the Imperial-based, marketing size, and the metric-based ETRTO size for each tire considered herein, where double-apostrophes (\") signify inches.  
Tire inflation pressures can be converted as follows:  
\[1\ \text{psi}=0.06894757\ \text{bar}. \tag{1}\]