applied sciences

Article Characterization and Modelling of Various Sized and the Effects of Tread Knobs and Inflation Pressure

Andrew Dressel 1,* and James Sadauckas 2 1 Departments of Mechanical & Civil Engineering, University of Wisconsin-Milwaukee, Milwaukee, WI 53211, USA 2 & Simulation Group, Harley-Davidson Motor Company, Wauwatosa, WI 53222, USA; [email protected] * Correspondence: [email protected]

 Received: 11 April 2020; Accepted: 29 April 2020; Published: 1 May 2020 

Abstract: Mountain bikes continue to be the largest segment of U.S. sales, totaling some USD 577.5 million in 2017 alone. One of the distinguishing features of the mountain bike is relatively wide tires with thick, knobby treads. Although some work has been done on characterizing street and commuter bicycle tires, little or no data have been published on off- tires. This work presents laboratory measurements of inflated tire profiles, tire footprints, and force and moment data, as well as static lateral and radial stiffness for various modern mountain bike tire sizes including plus size and fat bike tires. Pacejka’s Motorcycle Magic Formula tire model was applied and used to compare results. A basic model of tire lateral stiffness incorporating individual tread knobs as springs in parallel with the overall tread and the inflated carcass as springs in series was derived. Finally, the influence of inflation pressure was also examined. Results demonstrated appreciable differences in tire performance between 29 2.3”, 27.5 2.8”, 29 3”, and 26 4” knobby tires. × × × × The proposed simple model to combine tread knob and carcass stiffness offered a good approximation, whereas inflation pressure had a strong effect on mountain bike tire behavior.

Keywords: bicycle; mountain bike; pattern; force and moment; e-bike; tyre; dynamics

1. Introduction 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 . Tire durability is important because a flat tire can ruin a ride. Ride comfort, gleaned from the tire deflection, is a consideration even for 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.

Appl. Sci. 2020, 10, 3156; doi:10.3390/app10093156 www.mdpi.com/journal/applsci Appl. Sci. 2020, 10, 3156 2 of 22

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]. Pacejka’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-treaded 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. Table1 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 psi = 0.06894757 . (1)

Table 1. Imperial-based marketing size versus European Tyre and Rim Technical Organisation (ETRTO) designation.

Tire Size (in) ETRTO (mm) 29 2.3” 58–622 × 29 2.5” 63–622 × 29 3.0” 76–622 × 27.5 2.8” 71–584 × 26 4.0” 102–559 ×

2. Materials and Methods A brief description of the types of measurement conducted as well as the tools used for these measurements is followed by a discussion of data analysis methods prior to examining the results.

2.1. Measurement

2.1.1. Tire Inflated Radius Tire inflated radius was simply measured from ground to center of axle with the in a vertical upright position using a rule or measuring tape. This value defines the tire radius from axle to the crown, that is, the outermost point on the circumference of the undeformed, inflated tire profile.

2.1.2. Tire Profile Although various means exist in the automotive and motorcycle industry to measure tire profiles including laser scanning, coordinate measuring machine (CMM) arms, or other optical methods, this study chose a lower-tech, portable solution—a common machinist’s or carpenter’s contour gauge. Appl. Sci. 2020, 10, 3156 3 of 22

Appl.methods, Sci. 2020 this, 10 ,study 3156 chose a lower-tech, portable solution—a common machinist’s or carpenter’s3 of 22 contour gauge. As shown in Figure 1, this device consisted of a series of pins captured between two plates. As shown in Figure1, this device consisted of a series of pins captured between two plates. When the pins were pressed axially against an object, they slid between the plates with a small When the pins were pressed axially against an object, they slid between the plates with a small amount amount of frictional resistance, their tips conforming to the contour of the object. The user simply of frictional resistance, their tips conforming to the contour of the object. The user simply selects a selects a contour gauge of adequate width and depth, extends and levels the pins toward the object contour gauge of adequate width and depth, extends and levels the pins toward the object intended intended for measurement, and then slowly presses the device onto the object until the desired for measurement, and then slowly presses the device onto the object until the desired section shape section shape is captured. In the case of tire profiles, this process was conducted radially toward the is captured. In the case of tire profiles, this process was conducted radially toward the wheel center wheel center to capture the tire crown to shoulder (outboard edge of tread) shape across its width, to capture the tire crown to shoulder (outboard edge of tread) shape across its width, and was then and was then repeated orthogonally to the wheel plane to capture the side-wall contour, tire height, repeated orthogonally to the wheel plane to capture the side-wall contour, tire height, and rim interface. and rim interface. In the case of staggered knob arrangements, the process can be repeated for each In the case of staggered knob arrangements, the process can be repeated for each set of knobs. set of knobs.

Figure 1. Capturing mountain bike tire tread profileprofile with carpenter’s contour gauge.

The more complicated portion of the process was then digitizing and arranging the gauge results to accuratelyThe more represent complicated the tire’s portion geometry. of the process Following was each then gauge digitizing measurement, and arranging the gauge the gauge was aligned results to,to accurately and overlaid represent on, a piece the oftire graph’s geometry. paper and Following scanned. each The imagesgauge weremeasurement, then re-oriented, the gauge aligned, was andaligned scaled to, (ifand necessary) overlaid inon, a a photo piece processingof graph paper software and package.scanned. Finally,The images a MATLAB were then script re-oriented, was used toaligned, identify and the scaled edge (if of thenecessary) profile, in trace a photo it, and processing scale the resultsoftware with package. respect toFinally, the grid a MATLAB on the paper. script was used to identify the edge of the profile, trace it, and scale the result with respect to the grid on 2.1.3.the paper. Footprints Footprints were collected by coating the tire surface containing the expected contact patch with ink 2.1.3. Footprints using an office ink pad. The tire, which had been set to a specific inflation pressure, was then allowed to restFootprints vertically were on a collected piece of whiteby coating cardstock the tire paper surface with containing prescribed the normal expected load contact applied patch through with addedink using weights. an office In this ink case,pad. theThe force tire, andwhich moment had been fixture setwas to a used specific to accomplish inflation pressure, the task; was however, then aallowed bicycle to with rest rear vertically wheel on mounted a piece inof awhite stationary cardstock trainer paper and with over prescribed whose front normal wheel load appropriate applied weightsthrough wereadded applied weights. could In this also case, be used. the force and moment fixture was used to accomplish the task; however, a bicycle with rear wheel mounted in a stationary trainer and over whose front wheel 2.1.4.appropriate Force andweights Moment were applied could also be used. Force and moment measurements were performed with the tire test device at the University of Wisconsin–Milwaukee.2.1.4. Force and Moment As shown in Figure2 it consisted of a welded frame and an aluminum fork toForce hold and a bicycle moment wheel measurements in a desired orientationwere performed on top with of a small the tire treadmill test device of flat-top at thechain. University It had of a two-degreesWisconsin–Milwaukee. of freedom pivot,As shown implemented in Figure with2 it co annsisted automobile of a welded universal steel joint, frame far and (1.3 m)an forwardaluminum of thefork bicycle to hold tire a bicycle so that slightwheel variations in a desired in vertical orientation or horizontal on top of position a small produced treadmill negligible of flat-top variations chain. It inhad orientation a two-degrees angle. of The freedom forward pivot, pivot implemented was implemented with withan automobile needle-bearings universal so that joint, any far friction (1.3 m) in theforward bearings of the or sealsbicycle generated tire so athat negligible slight lateralvariations force in at vertical the contact or patch.horizontal position produced Appl. Sci. 2020, 10, 3156 4 of 22 negligible variations in orientation angle. The forward pivot was implemented with needle-bearings so that any friction in the bearings or seals generated a negligible lateral force at the contact patch.

This device allowed for sweeping and camber angles while measuring the lateral force, , and vertical moment, , generated in the contact patch. It used a force sensor to maintain the lateral location of the contact patch and a second force sensor acting on a lever arm of known length to prevent rotation of the fork that held the about its steering axis. In order to allow for the inevitable flexibility of the test frame and the bicycle wheel, the slip orientation of the bicycle rim was measured with a pair of laser position sensors mounted rigidly to the support platen for the flat-top chain near each end of the contact patch. Similarly, the camber orientation of the rim was measured with an accelerometer on the fork. was altered by pivoting the treadmill about a vertical axis under the center of the contact patch. was altered by tilting the fixture about the longitudinal axis of the universal joint, which passed through the contact patch. The vertical load borne by the tire was generated simply by the weight of the device’s frame. Additional mass could be added above the fork as desired. For this study, the applied normal load wasAppl. set Sci. 2020to 418, 10 ,N, 3156 which equated to front tire normal load of a 95 kg rider sitting on an 11 kg bicycle4 of 22 with 40% front and 60% rear weight distribution.

Figure 2. Tire force and moment measuring device at the University of Wisconsin–Milwaukee. Figure 2. Tire force and moment measuring device at the University of Wisconsin–Milwaukee.

TheThis lateral device force allowed generated for sweeping in the contact slip and patch camber was transmitted angles while through measuring the bicycle the lateral wheel force, to theFy, fork, and thus to the test device frame. From the frame, the force was transmitted to both the lateral and vertical moment, Mz, generated in the contact patch. It used a force sensor to maintain the lateral forcelocation sensor of the and contact the universal patch and joint. a second A simple force sensorstatic summing acting on aof lever the armmoments of known about length a vertical to prevent axis throughrotation ofthe the universal fork that joint held provided the bicycle a relationship wheel about between its steering the lateral axis. force generated in the contact patchIn and order the tolateral allow force for themeasured inevitable by the flexibility sensor. ofThe the mounting test frame point and on the the wheel, for the the lateral slip forceorientation sensorof was the positioned bicycle rim on was themeasured same axis withthroug a pairh the of center laser positionof the contact sensors patch mounted as the universal rigidly to jointthe support so that changes platen for in thecamber flat-top angle chain had nearno effect each on end the of lateral the contact force sensor patch. geometry. Similarly, the camber orientation of the rim was measured with an accelerometer on the fork. 2.1.5.Slip Static angle Lateral was and altered Radial by pivoting Stiffness the treadmill about a vertical axis under the center of the contact patch. Camber angle was altered by tilting the fixture about the longitudinal axis of the universal joint, which passed through the contact patch. The vertical load borne by the tire was generated simply by the weight of the device’s frame. Additional mass could be added above the fork as desired. For this study, the applied normal load was set to 418 N, which equated to front tire normal load of a 95 kg rider sitting on an 11 kg bicycle with 40% front and 60% rear weight distribution. The lateral force generated in the contact patch was transmitted through the bicycle wheel to the fork, and thus to the test device frame. From the frame, the force was transmitted to both the lateral force sensor and the universal joint. A simple static summing of the moments about a vertical axis through the universal joint provided a relationship between the lateral force generated in the contact patch and the lateral force measured by the sensor. The mounting point on the frame for the lateral force sensor was positioned on the same axis through the center of the contact patch as the universal joint so that changes in camber angle had no effect on the lateral force sensor geometry. Appl. Sci. 2020, 10, 3156 5 of 22

2.1.5. Static Lateral and Radial Stiffness Radial stiffness was measured by applying incremental weight to the top of the fork and measuring the resulting vertical deflection of the tire with a dial indicator. Static lateral stiffness was measured by pulling on the rim immediately above the contact patch with a force and simultaneously recording the force magnitude and the resulting lateral deflection of the rim. As illustrated in Section 3.2.4, the lateral shear stiffness of a specific number of knobs was measured in nearly the same way, except that the tire was not mounted on a wheel and the knobs were isolated from the rest of the tread by pressing them between two appropriately sized rectangular plates. The normal load was applied directly by the wheel and the test frame resting on the top rectangular plate. A lateral force was applied to the rim, as before, and the resulting lateral deflection was recorded. Non-skid tape, by 3M, was applied to the top of the treadmill to maximize friction for both the static lateral stiffness and the force and moment measurements.

2.2. Fitting the Data Pacejka’s Magic Formula was fit to the experimental force and moment data [6] in a two-step process. First the data was smoothed using a 1D weighted, Blaise filter. Then, coefficients for Pacejka’s Magic Formula, whose general form is depicted in Equation (2), were found to best fit the smoothed datasets for lateral force due to slip and camber, as well as to the self-aligning moment due to slip. Specifically, constraints from Pacejka’s Motorcycle Magic Formula Tire model, which more equally emphasizes the camber and slip angles, were applied. The utility of the Pacejka’s Motorcycle Magic Formula for this comparison were two-fold in that, given the specified constraints, each of the coefficients, and particular combinations thereof, had some physical significance, as explained below. Secondly, the fit data could be compared graphically and even extrapolated slightly to supplement the sometimes limited range of angles achieved during the physical test.  y = D sin Carctan[Bx E(Bx arctanBx)] , (2) − −

Y(X) = y(x) + SV, (3)

x = X + SH, (4) where

Y: output variable. In this case, either lateral force, Fy, or self-aligning moment, MZ ; • SA X: input variable; here, either slip or camber angle in radians; and • B: stiffness factor, which determined the slope at the origin; • C: shape factor, which controlled the limits of the range of the sine function; • D: peak value (when C was constrained as specified); • E: curvature factor for the peak, controlling its horizontal position; • SH: horizontal shift; • SV: vertical shift. • The product BCD, obtained by multiplying the respective Pacejka coefficients, corresponded to the slope at the origin, that is, linear stiffness of the data, which was normalized by applied vertical load, and represented the cornering stiffness coefficient for slip, camber stiffness coefficient, and self-aligning moment coefficient, respectively. Twisting torque due to camber, influenced predominantly by the difference in peripheral velocities across the ’s toroidal shape [7], was modelled using the nonlinear formula from the FastBike multibody simulation software depicted in Equation (5).

 2 MZTW (ϕ) = mrϕN 1 + twϕ , (5) Appl. Sci. 2020, 10, 3156 6 of 22 Appl. Sci. 2020, 10, 3156 6 of 22 where• : twisting torque due to camber, which was then normalized by • M:Z the: tire twisting normal torque load due in Newtons; to camber, which was then normalized by • TW N: the tire normal load in Newtons; and • and ϕ: camber angle in radians; •• : camber angle in radians; • mr: linear twisting torque coefficient; • : linear twisting torque coefficient; • tw: non-linear twisting torque coefficient. • : non-linear twisting torque coefficient. Figure3 shows the results of the curve fitting for the 29 2.3” knobby tire. The yellow dots Figure 3 shows the results of the curve fitting for the 29× × 2.3” knobby tire. The yellow dots representrepresent thethe cloudcloud ofof datadata collectedcollected duringduring thethe fixture’sfixture’s separateseparate sweepssweeps ofof slipslip andand camber.camber. TheThe thinthin blueblue lineline depictsdepicts the smoothed curve from the Blaise filter.filter. Finally, thethe redred dasheddashed lineline represents thethe respectiverespective fitfit curvecurve forfor thatthat data,data, whereaswhereas thethe redred plusplus symbolsymbol representsrepresents thethe originorigin ofof thethe fitfit curvecurve givengiven anyany ooffsets.ffsets. In this case, the Magic Formula curves for lateral force forfor bothboth slipslip andand cambercamber incorporatedincorporated aa smallsmall verticalvertical ooffset,ffset, whereaswhereas thatthat forfor self-aligningself-aligning momentmoment incorporatedincorporated bothboth aa smallsmall verticalvertical andand horizontalhorizontal ooffset.ffset. TheThe twistingtwisting torquetorque formulationformulation alsoalso containedcontained aa verticalvertical ooffsetffset term.term.

(a) (b)

(c) (d)

Figure 3. Force and moment raw data, smoothed lines, and fit curves for 29 × 2.3” knobby on 25 mm Figure 3. Force and moment raw data, smoothed lines, and fit curves for 29 2.3” knobby on 25 mm rim at 25 psi (1.7 bar) with (a) normalized lateral force vs. slip angle, (b) normalized× lateral force vs. rim at 25 psi (1.7 bar) 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 camber angle, (c) normalized self-aligning moment vs. slip angle and (d) normalized twisting torque vs.vs. camber angle.

As can be seen, the lateral force versus slip (Figure 3a) and, to an even greater extent, the self- aligning moment versus slip (Figure 3c) exhibited the characteristic shape of the Magic Formula, Appl. Sci. 2020, 10, 3156 7 of 22

As can be seen, the lateral force versus slip (Figure3a) and, to an even greater extent, the self- aligning moment versus slip (Figure3c) exhibited the characteristic shape of the Magic Formula, whereas the lateral force versus camber (Figure3b) showed less curvature, as did the twisting torque (Figure3d), with some negative (downward) curvature in this example. It should be noted that the plot scales were fixed to allow comparison across all datasets reported herein. In this case, although the fixture was capable of 30 degrees of camber, the 29 2.3” knobby was × only measured to about +15 and 25 degrees, whereas slip angles achieved by the fixture for this − particular tire were on the order of +/ 2 degrees. In contrast large, heavy-duty automotive fixtures − test slip angles up to, or in excess of, five or six degrees. Although such slip angles may be seen in aggressive transient or racing maneuvers, the added allure of measuring to these extremes is to better capture the peak and subsequent asymptote of the curves. In the case of the bicycle tire measurements and particularly the wider, knobby mountain bike tire measurements, the width of the treadmill, which was originally designed and built for testing relatively narrow road tires, was the limiting factor. As the wide tires were rotated in slip or in camber, care needed to be taken so that the tire did not encounter the platen that supported the treadmill. As such, the ability to accurately identify the curvature, peak, and asymptote terms for lateral force versus slip was limited and became even more limited with wider tires, which in the case of the 26x4” may only achieve slip angles up to +/ 1 degree with the − studied treadmill arrangement. Regardless, the slope of the respective curves near the origin, that is, the stiffness or product of BCD Pacejka coefficients, was well captured and proved useful for the subsequent comparisons herein.

3. Results and Discussion

3.1. Various Sizes of Mountain Bike Tires For decades, mountain bikes were equipped with so-called 26-inch . This was in reference to the approximate outer tire diameter, consisting of a rim with approximately a 22-inch (559 mm) outer diameter and a tire that is about 2 inches tall at top and bottom. The so-called “29er” wheel and tire size, with a 24.5-inch (622 mm) diameter rim (equal to that of “700c” road bike wheels) gained popularity in the early 2000s on the basis of its purported ability to roll over obstacles with greater ease. However, some riders still yearned for the quick handling of smaller diameter wheels. After significant tooling investments from the bicycle industry, the so-called 27.5-inch or “650b” wheel size, with about a 23-inch (584 mm) rim diameter, offered what some thought was the optimal compromise. As wheel diameters ebbed and flowed, pioneers within the industry have also explored various tire widths. Fat bike tires, which are generally 3.8 inches (96.5 mm) or wider, grew out of a desire to ride on soft snow and sand. The “plus tire”, 2.6 inches (66 mm) to roughly 3 inches (76 mm) wide, split that difference, allegedly trading some of the original 29er’s outright speed for more tire volume and confidence-inspiring . Now “29 plus” tires take an extreme to the extreme in terms of both diameter and width, and are a growing segment in both enduro mountain bike and bike packing applications. As tire sizes have evolved, new rim widths have been adapted to follow suit. In the past, 18 or 19 mm inner width rims, similar to those used on past road racing bikes, were the norm. Over the past decade, rims gradually grew in width, chasing various trade-offs and trends in wheel stiffness, tire stiffness, tire profile, weight, and strength. For this study, “modern” rim widths appropriate to each tire size were selected on the basis of realistic use case and/or market benchmarking within that segment. Inner rim width will be referred to in millimeters where appropriate. Similarly, each of the above tire and rim combinations have their own performance trade-offs in terms of “nominal” inflation pressure for a given rider weight, bike fitment, and terrain. Nominal inflation pressures used for each tire size in the following study are based on the author’s experience with these tire setups for summer off-road trail riding. Appl. Sci. 2020, 10, 3156 8 of 22

For the remainder of the paper, the 29 2.3” knobby tire on 25 mm inner rim width at 25 psi × inflationAppl. Sci. pressure 2020, 10, 3156 is considered the baseline to which data of the other tire configurations are compared.8 of 22

3.1.1.3.1.1. Cross-Sectional Cross-Sectional Profiles Profiles FigureFigure4 shows4 shows side-by-side side-by-side comparisons comparisons ofof measured,measured, inflated inflated tire tire profiles profiles for for four four different di fferent knobbyknobby mountain mountain bike bike tire tire sizes. sizes. All All the the tires tires were were from from the the same same manufacturer, manufacturer, andand thethe threethree tirestires on theon left the are left all are the all samethe same tire tire model model only only in in di differefferentnt sizes. sizes. The tire on on the the right right is isa different a different model. model. AsAs described described previously, previously, as theas treadthe tread pattern pattern often ofte containsn contains alternating alternating sets ofsets knobs of knobs at regular at regular intervals alongintervals the tire’s along circumference, the tire’s circumference, these profiles these overlay profiles each ofoverlay the individual each of knobthe arrangementsindividual knob onto arrangements onto one cross section. This was done to better understand and compare the cross- one cross section. This was done to better understand and compare the cross-sectional shapes, that is, sectional shapes, that is, the effective toroid radius of each tire, although it does potentially limit the the effective toroid radius of each tire, although it does potentially limit the ability to visually decipher ability to visually decipher spacing between individual knobs, which can be read from the footprints spacing between individual knobs, which can be read from the footprints instead. instead.

Figure 4.FigureCross 4. sectionalCross sectional profiles profiles for 29 for2.3”, 29 × 27.5 2.3”, 27.52.8”, × 29 2.8”,3”, 29 and× 3”, 26 and4” 26 knobby × 4” knobby tires, tires, respectively. × respectively.× × × The tires shown increase in width from left to right. In terms of inflated outer radii (or diameters), the 29 The3” istires by farshown the largest,increase followed in width by from the 29 left 2.3”,to right. which In is terms very slightlyof inflated larger outer than radii the 26(or 4” × × × fatdiameters), bike tire, and the finally29 × 3” the is by 27.5 far the2.8” largest, plus tire. followed by the 29 × 2.3”, which is very slightly larger than the 26 × 4” fat bike tire, and× finally the 27.5 × 2.8” plus tire. The shape of these mountain bike tires’ inflated profiles is obviously influenced by rim width, The shape of these mountain bike tires’ inflated profiles is obviously influenced by rim width, as can be seen in the curvature of the tires’ sidewalls and the angle that the lower sidewalls assume as can be seen in the curvature of the tires’ sidewalls and the angle that the lower sidewalls assume toward their respective bead interfaces. Here, rim width and rim-width-relative-to-tire-width increase toward their respective bead interfaces. Here, rim width and rim-width-relative-to-tire-width from left to right. increase from left to right. ExaminingExamining the the treaded treaded portion portion of of each each profile, profile, intended intended to to interact interact with with the the ground ground as as the the tire tire rolls and deforms and as the bicycle cambers and steers, several observations can be made. The 29 2.3” rolls and deforms and as the bicycle cambers and steers, several observations can be made. The 29× × tire2.3” profile tire profile was rather was rather flat, that flat, is, that having is, having a large a la toroidalrge toroidal radius, radius, with with limited limited drop drop from from the twothe two center rowscenter of knobs rows toof theknobs shoulder to the knobs. shoulder It is knobs. also worth It is notingalso worth that thenoting knob that widths the knob were relativelywidths were small, andrelatively the shoulder small, knobs and the were shoulder almost knobs equal were in height almost and equal width in height but with and considerablewidth but with draft considerable (or bracing) downdraft toward (or bracing) the tire down sidewall toward to presumablythe tire sidewall support to presumably the knob. The support middle the two knob. profiles The middle represent two the “plus”profiles tires represent with the the 27.5 “plus”2.8” tires on with the leftthe 27.5 and × 29 2.8” 3”on onthe the left right. and 29 These × 3” on tires the hadright. a similarThese tires tread × × patternhad a tosimilar the 29tread2.3”, pattern but to both the 29 knob × 2.3”, width but both and spacingknob width increased and spacing as the increased tire size as increased. the tire size It is × uncertainincreased. if thisIt is treaduncertain scaling if this is for tread aesthetic scaling or is performancefor aesthetic or reasons. performance Notice reasons. the large Notice gap between the large the centergap rowsbetween of knobsthe center and shoulderrows of knobs knobs and on theshoulder 29 3” knobs tire. on Notice the 29 also × 3” that tire. the Notice angle ofalso the that shoulder the × knobangle was of steeperthe shoulder and that knob (neglecting was steeper deformation) and that (neglecting the tire woulddeformation) need to the roll tire farther would for need the to tangent roll (groundfarther line) for the to engagetangent the(ground shoulder line) knobs. to engage The the 26 sh4”oulder fat bike knobs. tire The on the 26 right× 4” fat had bike a di tirefferent on the tread × patternright had with a sevendifferent rows tread of knobspattern as wi opposedth seven torows the of four knobs of theas opposed other tires, to the and four included of the other a center tires, row. and included a center row. 3.1.2. Footprints 3.1.2. Footprints As described in the various literature on cars [8] and motorcycles [9], tire footprints can tell us a lot aboutAs described the interaction in the various between literature the tire on and cars ground. [8] and motorcycles Normal load [9], is tire applied footprints to the can tire, tell us which a deformslot about to createthe interaction a contact patch,between or the footprint, tire and which ground. at zero Normal camber load angle is applied is quite to elliptical the tire, on which smooth, deforms to create a contact patch, or footprint, which at zero camber angle is quite elliptical on smooth, flat ground. The distribution of the normal load over the contact patch area yields the contact Appl. Sci. 2020, 10, 3156 9 of 22

Appl. Sci. 2020, 10, 3156 9 of 22 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 patch pressure (not measured here). In the absence of camber, the contact patch pressure may take fairly regular distribution. The presence of tread knobs obviously localizes contact pressure. Similarly, on a fairly regular distribution. The presence of tread knobs obviously localizes contact pressure. the knobs localize the shear used to generate lateral and longitudinal friction forces as the tire is Similarly, the knobs localize the shear stress used to generate lateral and longitudinal friction forces slipped, cambered, driven or braked. Although not the focus of this work, it has been suggested that as the tire is slipped, cambered, driven or braked. Although not the focus of this work, it has been even for car tires, depending on (more closely spaced) tread pattern, the localized pressure can be 10 suggested that even for car tires, depending on (more closely spaced) tread pattern, the localized× the average [10]. pressure can be 10x the average [10]. Figure5 presents the knobby mountain bike tire footprints in the same order and on the same 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 Figure4. Here, the fitted ellipse was superimposed on the footprint 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, and its axes were used to calculate length and width of the contact patch. Examining overall it can be seen that the 26 4” fat bike tire had both the longest and widest contact patch followed in dimensions, it can be seen× that the 26 × 4” fat bike tire had both the longest and widest contact patch length by the 29 3” plus tire, the 29 2.3” knobby, and 27.5 2.8”. Hence, the length of the contact followed in length× by the 29 × 3” plus× tire, the 29 × 2.3” knobby,× and 27.5 × 2.8”. Hence, the length of patch depended not only on tire outer diameter, but presumably also on deflection, as influenced by the contact patch depended not only on tire outer diameter, but presumably also on deflection, as inflation pressure, vertical load, and the tire’s carcass. In terms of contact patch width, the 26 4” was influenced by inflation pressure, vertical load, and the tire’s carcass. In terms of contact patch× width, widest, whereas 27.5 2.8” and 29 3” were essentially equal, despite their size designation, and the the 26 × 4” was widest,× whereas 27.5× × 2.8” and 29 × 3” were essentially equal, despite their size 29 2.3” at its nominal inflation pressure was narrower. designation,× and the 29 × 2.3” at its nominal inflation pressure was narrower.

FigureFigure 5.5. FootprintsFootprints forfor 2929 × 2.3”, 27.5 × 2.8”, 29 × 3”,3”, and and 26 26 × 4”4” knobby knobby tires, tires, respectively. respectively. × × × × These images bring very clearclear meaningmeaning toto thethe tiretire industryindustry termterm “void“void ratio”ratio” [[11].11]. For thesethese knobby tires,tires, the the amount amount of white,of wh emptyite, empty space space within within the contact the contact patch ellipse patch considerably ellipse considerably exceeded theexceeded amount the of amount black tread of black interacting tread interacting with ground. with The ground. same The MATLAB same scriptMATLAB used script for fitting used thefor ellipsefitting the also ellipse computed also computed the void ratio the byvoid comparing ratio by comparing white space white to contact space patch to contact area afterpatch converting area after theconverting image to the binary. image The to binary. resulting The void resulting ratios in void terms ratios of percentage in terms of for percentage each knobby for each tire areknobby listed tire on theare figurelisted on and the ranged figure from and 69.7%ranged to from 80.6%. 69.7% to 80.6%. As noted in thethe cross-sectionalcross-sectional profiles,profiles, thethe treadtread patternpattern ofof thethe 27.527.5 × 2.8” andand 2929 × 3” tires × × essentially scaled scaled the the size size and and spacing spacing of of the the 29 29× 2.3”2.3” tread tread pattern pattern upward upward in proportion in proportion to tire to size. tire × size.Again, Again, the tread-pattern the tread-pattern of these of these three three tires tiresoffered offered no center no center ridge ridge of knobs, of knobs, whereas whereas the 26 the × 264” tire4” × tiredid. did. Note that the three tread patterns on the left em employedployed essentially essentially two two different different knob typologies, typologies, one thatthat was was rectangular rectangular and oneand thatone was that slightly was wedge-shapedslightly wedge-shaped with an hourglass-shaped with an hourglass-shaped longitudinal groovelongitudinal whose groove depth waswhose roughly depth half was of roughly the 4.5mm half tread of the depth. 4.5 mm The tread 26 depth.4” fat bike Thetire, 26 × on 4” the fatright, bike × showedtire, on fourthe right, knob typologiesshowed four within knob its typologies footprint, twowithin alternating its footprint, central two knob alternating chevrons withcentral varying knob degreeschevrons of with central varying and trailing-edge degrees of relief,central and and two trai typesling-edge of intermediate relief, and knobs two types similar of to intermediate those of the otherknobs tires. similar Note to those that at of zero the camberother tires. angle Note and that the inflationat zero camber pressures angle shown, and the the inflation footprints pressures did not engageshown, the footprints shoulder knobs,did not nor engage the more the shoulder outboard kn bandobs, nor of intermediatethe more outboard knobs band on the of 26x4”intermediate fat tire knobs on the 26x4” fat tire shown in the profiles of Figure 4. It should be noted that purposefully Appl. Sci. 2020, 10, 3156 10 of 22

shownAppl. in Sci. the 2020 profiles, 10, 3156 of Figure4. It should be noted that purposefully positioning the wheel’s rotation10 of 22 during footprint tests to engage leading and trailing knobs to better define borders of the contact patch greatlypositioning assisted the in thewheel’s identification rotation during of the footprint ellipse. tests to engage leading and trailing knobs to better defineFinally, borders these of images the contact of the patch footprint greatly wereassisted useful in the in identification identifying of how the ellipse. many knobs, pairs of knobs, orFinally, partial these knobs images are of in the contact footprint at awere given useful inflation in identifying pressure. how Thismany becomes knobs, pairs relevant of knobs, in a subsequentor partial section. knobs are in contact at a given inflation pressure. This becomes relevant in a subsequent section. 3.1.3. Forces and Moments 3.1.3. Forces and Moments Force and moment plots based on the Pacejka Magic coefficients and FastBike fit parameters are shown inForce Figure and6. moment Although plots each based dataset on the retained Pacejka a Magi specificc coefficients marker type, and FastBike its line stylefit parameters was divided are intoshown two sections. in Figure The6. Although solid line each portion, dataset emanating retained a specific from the marker origin, type, indicates its line valuesstyle was within divided the measurementinto two sections. range, whereasThe solid the line dashed portion, portion emanating of the from line representsthe origin, anyindicates extrapolated values within values the on the basismeasurement of the fit. range, As described whereas previously,the dashed theportion limited of the treadmill line represents width of any the extrapolated test fixture constrained values on slipthe and basis camber of the ranges, fit. As described particularly previously, for thewider the limited knobby treadmill tires. width Although of the the test linear fixture portion constrained of the 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 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 coefficients, was well identified, the curvature toward the peak and the eventual asymptote often had lower resolution, as was evident when comparing the 29 2.3” solid versus dashed line to that of the lower resolution, as was evident when comparing the 29× × 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 26x4”. As such, caution should be exercised if attempting to examine peak or curvature behavior of the widerthe wider tire tire variants, variants, especially especially in in terms terms of of slip slip angle. angle.

(a) (b)

(c) (d)

FigureFigure 6. Force 6. Force and and moment moment fitted fitted curves curves forfor 2929 × 2.3”,2.3”, 27.5 27.5 × 2.8”,2.8”, 29 × 29 3”, and3”, 26 and × 4” 26 knobby4” knobby tires × × × × tires withwith ( a(a)) normalized normalized lateral force force vs. vs. slip slip angle, angle, (b) (normalizedb) normalized lateral lateral force force vs. camber vs. camber angle, angle,(c) (c) normalizednormalized self-aligning self-aligning moment moment vs. slipvs. slip angle angle and and (d) ( normalizedd) normalized twisting twisting torque torque vs. vs. camber camber angle. angle. Appl. Sci. 2020, 10, 3156 11 of 22

Upon examining each of the plots in Figure6, several interesting trends were evident. In Figure6a, the upper left plot of normalized lateral force versus sideslip, the 29 2.3” had the lowest cornering × stiffness. The 27.5 2.8”and 29 3” plus tires shared similar, steeper slopes but differed in curvature, × × whereas the 26 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 (Figure6b), 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

Fy = tan(ϕ), (6) N where Fy is the lateral force, N the normal load on the tire, and ϕ the camber angle. Any tire below this line, such as the 29 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 (Figure6c) 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 , 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 Figure7, 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. The shape of the pneumatic trail curve is sometimes fit with a cosine function in some versions of Pacejka’s Magic Formula. As can been seen, the tire with the longest contact patch, in this case the 26 4”, does not necessarily have the largest value of pneumatic trail near zero-slip, however, × the other tires did follow that trend. It is possible that impending curvature of the 26 4” self-aligning × moment was not captured due the limited slip angle measurement range for that tire. In this case, the tires with a wider contact patch with respect to their length had a pneumatic trail that decayed more slowly. The lower right plot in Figure6d shows normalized tire twisting torque versus camber. This torque was due to fore aft shear stress distribution across the contact patch width driven by the toroidal tire shape and acted to steer the wheel into the corner in the direction of lean. Here, the 26 4” fat bike × tire still dominated, followed by the 27.5 2.8” plus tire, whereas at low camber angles the 29 2.3” × × tire trumped the 29 3”. It is interesting to note that at higher camber angles (or lower inflation × pressures, shown later), the 29 3” plus tire’s twisting torque increased to a level just beyond that × of the 27.5 2.8”. It was hypothesized that as the wider-spaced shoulder knobs of the 29 3” tire × × encounter ground, they effectively increase the width of the patch thus augmenting the twisting torque. The high twisting torque of the 26 4” fat bike tires seemed to corroborate the anecdotal depictions of × heavy feeling, “autosteer” on many fat bikes as they were leaned into a corner, which requires the rider to counteract an increase in steering angle with significant steering effort. Appl. Sci. 2020, 10, 3156 11 of 22

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 × 2.3” had the lowest cornering stiffness. The 27.5 × 2.8”and 29 × 3” plus tires shared similar, steeper slopes but differed in curvature, whereas the 26 × 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 =tan() , (6)

where is the lateral force, N the normal load on the tire, and the camber angle. Any tire below this line, such as the 29 × 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. The shape of the pneumatic trail curve is sometimes fit with a cosine function in some versions of Pacejka’s Magic Formula. As can been seen, the tire with the longest contact patch, in this case the 26 × 4”, does not necessarily have the largest value of pneumatic trail near zero-slip, however, the other tires did follow that trend. It is possible that impending curvature of the 26 × 4” self-aligning moment was not captured due the limited slip angle measurement range for that tire. In this case, the tires with a wider contact patch with respect to their length had a pneumatic trail that Appl. Sci. 10 decayed2020 more, , slowly. 3156 12 of 22

Appl. Sci. 2020, 10, 3156 12 of 22

The lower right plot in Figure 6d shows normalized tire twisting torque versus camber. This torque was due to fore aft shear stress distribution across the contact patch width driven by the toroidal tire shape and acted to steer the wheel into the corner in the direction of lean. Here, the 26 × 4” fat bike tire still dominated, followed by the 27.5 × 2.8” plus tire, whereas at low camber angles the 29 × 2.3” tire trumped the 29 × 3”. It is interesting to note that at higher camber angles (or lower inflation pressures, shown later), the 29 × 3” plus tire’s twisting torque increased to a level just beyond that of the 27.5 × 2.8”. It was hypothesized that as the wider-spaced shoulder knobs of the 29 × 3” tire encounter ground, they effectively increase the width of the patch thus augmenting the twisting torque. The high twisting torque of the 26 × 4” fat bike tires seemed to corroborate the anecdotal depictions of heavy feeling, “autosteer” on many fat bikes as they were leaned into a corner, which requires theFigureFigure rider 7. 7. Calculated toCalculated counteract pneumaticpneumatic an increase trailtrail forfor in 2929steering × 2.3”,2.3”, 27.5angle27.5 × 2.8”,with 29 significant × 3”,3”, and and 26 26steering × 4”4” knobby knobby effort. tires. tires. × × × × 3.1.4.3.1.4. Static Static Lateral Lateral and and Radial Radial Stiffness Stiffness AlthoughAlthough the the static static lateral lateral and and radial radial stistiffnessffness testtest waswas fairly fairly straightforward straightforward and and did did not not take take into intoaccount account any any stiff stiffeningening of the of carcassthe carcass due due to centrifugal to centrifugal effects effects of the of tire the mass tire beingmass being accelerated accelerated radially radiallyas the wheelas the spins,wheel thesespins, arethese still are very still useful very useful parameters parameters for understanding for understanding and modelling and modelling bicycle bicycleout of out plane of plane stability, stability, in the in case the ofcase lateral of latera stiffness,l stiffness, and in-plane and in-plane compliance, compliance, in the in case the ofcase radial of radialstiffness. stiffness. Here, Here, it seemed it seemed that that for thefor the given given rim rim width width and and “nominal” “nominal” inflation inflation pressure pressure selected selected for foreach each size, size, the the lateral lateral sti ffstiffness,ness, shown shown in Figurein Figure8a, increased8a, increased with with tire (andtire (and/or/or rim) rim) width width and theand radial the radialstiffness, stiffness, shown shown in Figure in Figure8b for 8b all for four all configurations,four configurations, happened happened to converge to converge around around 60,000 60,000 N /m. N/m.The The maximum maximum value value of the of vertical the vertical axis on axis these on plotsthese was plots set was to roughly set to roughly the lower the bound lower of bound stiffnesses of stiffnessesfor motorcycle for motorcycle tires [7]. Hence, tires bicycle[7]. Hence, tires at bicy thecle inflation tires pressuresat the inflation shown werepressures significantly shown lesswere sti ff significantlythan motorcycle less stiff tires. than motorcycle tires.

(a) (b)

FigureFigure 8. 8.(a)( aStatic) Static lateral lateral and and (b ()b radial) radial stiffness stiffness of of 29 29 × 2.3”,2.3”, 27.5 27.5 × 2.8”,2.8”, 29 29 × 3”,3”, and and 26 26 × 4”4” knobby knobby × × × × bicyclebicycle tires. tires.

3.2.3.2. Effect Effect of ofTread Tread Knobs Knobs InIn order order to to examine examine the the effect effect of of tread tread knobs, knobs, this this study study investigated investigated two two levels levels of ofresolution. resolution. First,First, a aless less treaded treaded tire tire with similar cross-sectionalcross-sectional profileprofile and and undeformed, undeformed, inflated inflated outer outer radius radius was measured for comparison to the 29 2.3” knobby tire. Although this tire appeared much smoother was measured for comparison to the× 29 × 2.3” knobby tire. Although this tire appeared much smootherthan the than knobby, the itknobby, did indeed it did possess indeed a possess “file-tread” a “file-tread” pattern consisting pattern consisting of many small,of many pyramidal, small, pyramidal,and closely and spaced closely knobs spaced (each knobs roughly (each roughly 1.5 mm 1.5 wide mm at wide their at base their and base equally and equally tall at tall their at their peak). peak). The lesser knob height required a slightly larger tire size of 29 × 2.5” to achieve a similar outer diameter, as evident in the tire cross-sectional profiles. After characterization of both the knobby baseline and file-tread tires, the second level of resolution involved sanding off the tread of both tires and re-characterizing what are referred to herein as the “bald” variants, as shown in Figure 9. In this case, the knobby tire became something like a true “slick” per Figure 9a, whereas the negative grooves of the file-tread in the Figure 9b tire remained. The various colors that appeared in the bald 29 × 2.3” where the knobs previously existed Appl. Sci. 2020, 10, 3156 13 of 22

The lesser knob height required a slightly larger tire size of 29 2.5” to achieve a similar outer diameter, × as evident in the tire cross-sectional profiles. After characterization of both the knobby baseline and file-tread tires, the second level of resolution involved sanding off the tread of both tires and re-characterizing what are referred to herein as the “bald” variants, as shown in Figure9. In this case, the knobby tire became something like a true “slick” perAppl. Figure Sci. 20209a,, 10 whereas, 3156 the negative grooves of the file-tread in the Figure9b tire remained. The various13 of 22 colorsAppl. Sci. that 2020 appeared, 10, 3156 in the bald 29 2.3” where the knobs previously existed suggested di13ff erentof 22 × suggestedcompounds different used to compounds form the carcass used andto form the knobs.the carcass The and results the areknobs. presented The results in the are same presented format asin suggested different compounds used to form the carcass and the knobs. The results are presented in the same previous format section. as the previous section. the same format as the previous section.

((aa)) ((bb))

FigureFigure 9. 9. Images Images of of of ( (a (a)a) )original original original treadedtreaded treaded 2929 29 ×× 2.3”2.3” knobby knobby and and itsits baldbald its bald tiretire counterpart tirecounterpart counterpart and and ( andb(b) )the the (b 29) 29 the × × × 2.5”292.5” file-tread 2.5”file-tread file-tread tire tire with with tire with itsits baldbald its baldtiretire counterpart.counterpart. tire counterpart. × 3.2.1. Cross-Sectional Profiles 3.2.1.3.2.1. Cross-Sectional Cross-Sectional Profiles Profiles Figure 10 again shows the cross-sectional profile of the baseline 29 2.3” knobby tire, this time FigureFigure 10 10 again again showsshows thethe cross-sectionalcross-sectional profile of thethe baselinebaseline 2929 ×× 2.3”2.3” knobbyknobby tire, tire, this this time time with the 29 2.3” “bald” variant overlaid. As captured by the measurement, all tread knobs were withwith the the 29 29 ×× × 2.3”2.3” “bald”“bald” variantvariant overlaid.overlaid. As captured by the measurement,measurement, allall treadtread knobs knobs were were completely removed, and this was done around the tire’s entire circumference. Similarly, the 29 2.5” completelycompletely removed, removed, andand thisthis waswas donedone around the tire’s entireentire circumference.circumference. Similarly,Similarly, the the ×29 29 × × file-tread and 29 2.5” bald tire are shown in the overlay. Here, the removal of the more tightly spaced 2.5”2.5” file-tread file-tread and and× 29 29 ×× 2.5”2.5” baldbald tiretire areare shown in the overlay. Here,Here, thethe removalremoval of of the the more more tightly tightly spacedandspaced shorter and and heightshorter shorter knobs height height resulted knobsknobs resultedresulted in a more in subtle, a more annular subtle, reductionannular reductionreduction in the treaded inin the theportion treaded treaded of portion portion the tire ofcrossof the the section.tire tire cross cross section. section.

Figure 10. Cross sectional profiles of 29 × 2.3” knobby tire (solid) overlaid with its bald variant Figure 10.10. CrossCross sectional sectional profiles profiles of 29of 292.3” × 2.3” knobby knobby tire (solid) tire (solid) overlaid overlaid with its with bald variantits bald (dashed), variant (dashed), on the left, and 29 × 2.5” file-tread× tire (solid) overlaid with its bald variant (dashed), on the (dashed),on the left, on and the 29 left, 2.5”and file-tread29 × 2.5” file-tread tire (solid) tire overlaid (solid) withoverlaid its bald with variant its bald (dashed), variant (dashed), on the right. on the right.right. × 3.2.2. Footprints 3.2.2. Footprints 3.2.2.Figure Footprints 11 shows tire footprints for the 29 2.3” baseline knobby tire and bald variant, as well × as theFigureFigure 29 2.5”11 11 shows shows file-tread tire tire footprintsfootprints and bald for variant,for the 29 all × 2.3” on 25 baseline mm width knobby rims tiretire at andand 25 psi baldbald (1.7 variant, variant, bar). as Theas well well small, as as × thethe 29 29 × × 2.5” 2.5” file-tread file-tread and and baldbald variant,variant, allall onon 25 mm width rimsrims atat 2525 psipsi (1.7(1.7 bar). bar). The The small, small, closely closely spacedspaced file-tread file-tread knobs, knobs, asas wellwell asas thethe negativenegative tread grooves onon thethe 2929 ×× 2.5”,2.5”, cancan be be seen. seen. The The file- file- treadtread was was effectively effectively removedremoved fromfrom thethe 2929 × 2.5” bald variant, butbut thethe negativenegative groovesgrooves remained. remained. TheThe 29 29 × × 2.3” 2.3” bald bald tire tire waswas essentiallyessentially aa slick. Appl. Sci. 2020, 10, 3156 14 of 22 closely spaced file-tread knobs, as well as the negative tread grooves on the 29 2.5”, can be seen. × The file-tread was effectively removed from the 29 2.5” bald variant, but the negative grooves × remained. The 29 2.3” bald tire was essentially a slick. Appl. Sci. 2020, 10, 3156× 14 of 22

Figure 11. Footprints for 29 × 2.3” knobby, 29 × 2.3” bald, 29 × 2.5” file-tread, and 29 × 2.5” bald tires Figure 11. Footprints for 29 2.3” knobby, 29 2.3” bald, 29 2.5” file-tread, and 29 2.5” bald tires at 25 psi (1.7 bar) on 25 mm× inner width rim. × × × at 25 psi (1.7 bar) on 25 mm inner width rim. The 29 × 2.3” bald contact patch was both shorter and narrower than that of the 29 × 2.3” knobby The 29 2.3” bald contact patch was both shorter and narrower than that of the 29 2.3” knobby tire, whereas× the 29 × 2.5” bald contact patch was longer but narrower than that of the 29× × 2.5” file- tire, whereas the 29 2.5” bald contact patch was longer but narrower than that of the 29 2.5” tread tire. It is interesting× to note that, despite having a smaller outer radius than the 29 × 2.5” tires,× file-tread tire. It is interesting to note that, despite having a smaller outer radius than the 29 2.5” tires, the 29 × 2.3” knobby tire yielded the widest and longest contact patch of these four variants.× the 29 2.3” knobby tire yielded the widest and longest contact patch of these four variants. The× relatively high (66.4%) void ratio of the 29 × 2.5” file-tread might seem surprising, as this The relatively high (66.4%) void ratio of the 29 2.5” file-tread might seem surprising, as this tire tire was chosen as a less-treaded comparison to the× 29 × 2.3” knobby (78.1%). Even after removing was chosen as a less-treaded comparison to the 29 2.3” knobby (78.1%). Even after removing the the file-tread pattern, the remaining negative tread grooves× yielded an appreciable (19.6%) void ratio file-treadfor the 29 pattern, × 2.5” thebald remaining tire, as opposed negative to tread the groovesnearly slick yielded (2.2%) an appreciablevoid ratio of (19.6%) its 29 void× 2.3” ratio bald for the 29 2.5” bald tire, as opposed to the nearly slick (2.2%) void ratio of its 29 2.3” bald counterpart. counterpart.× ×

3.2.3.3.2.3. Forces Forces and and MomentsMoments VariousVarious trends trends are are evident evident inin thethe fittedfitted forceforce and moment plots plots depicted depicted in in Figure Figure 12. 12 The. The upper upper leftleft plot plot of of normalized normalized lateral lateral force force versus versus sideslip sideslip (Figure (Figure 12a) shows12a) shows an increase an increase in cornering in cornering stiffness as the tires became less treaded. There was a large increase in slope from the 29 2.3” knobby to the stiffness as the tires became less treaded. There was a large increase in slope from the× 29 × 2.3” knobby 29 2.3” bald tire, whereas the 29 2.5” file-tread to bald modification showed an increase but of to× the 29 × 2.3” bald tire, whereas the× 29 × 2.5” file-tread to bald modification showed an increase but lessof less magnitude. magnitude. TheThe same same pattern pattern was was repeatedrepeated inin thethe other three plots. Removing Removing the the knobs knobs had had a abig big effect effect and and removingremoving the the file-tread file-tread had had a smalla small eff effect,ect, both both trending trending in similarin similar directions. directions. Overall, Overall, the the slick slick tire tire was thewas sti theffest, stiffest, and the and knobby the knobby tire was tire the was least the sti leastff. As stiff. noted As noted previously, previously, the shape the ofshape the curveof the beyondcurve thebeyond recorded the data,recorded indicated data, indicated by the dashed by the lines, dashed was lines, an extrapolation was an extrapolation based on thebased fit coeon ffithecients. fit Thus,coefficients. the curvature Thus, the of somecurvature of the of more some limited of the mo datasetsre limited may datasets have been may exaggerated. have been exaggerated.

Appl. Sci. 2020, 10, 3156 15 of 22

Appl. Sci. 2020, 10, 3156 15 of 22

(a) (b)

(c) (d)

FigureFigure 12.12. Force andand momentmoment fittedfitted curvescurves forfor 2929 × 2.3” knobby,knobby, 2929 × 2.3” bald, 2929 × 2.5” file-tread,file-tread, × × × andand 2929 × 2.5”2.5” baldbald tirestires at at 25 25 psi psi (1.7 (1.7 bar) bar) on on 25 25 mm mm inner inner width width rim rim with with (a) a) normalized normalized lateral lateral force force vs. × slipvs. slip angle, angle, (b) normalizedb) normalized lateral lateral force force vs. cambervs. camber angle, angle, (c) normalized c) normalized self-aligning self-aligning moment moment vs. slip vs. angleslip angle and (andd) normalized d) normalized twisting twisting torque torque vs. cambervs. camber angle. angle.

3.2.4.3.2.4. Static Lateral StiStiffnessffness ToTo understandunderstand thethe contributioncontribution ofof tiretire treadtread knobsknobs toto totaltotal (treaded)(treaded) tiretire laterallateral stistiffness,ffness, knobknob shearshear stistiffness,ffness, andand baldbald carcasscarcass laterallateral stistiffnessffness werewere measuredmeasured separatelyseparately andand thenthen combined,combined, inin thethe mannermanner ofof springssprings in in series, series, as as in in Equation Equation (7), (7), to to compare compare against against the the measured measured lateral lateral stiff stiffnessness of the of knobbythe knobby tire. tire. 1 1 1 = + , (7) K =K + K , (7) total tire bald carcass tread AsAs illustratedillustrated in FigureFigure 1313,, thethe shearshear stistiffnessffness ofof thethe treadtread knobsknobs waswas measuredmeasured byby isolatingisolating anan integralintegral numbernumber ofof knobsknobs (six(six shown)shown) ofof anan unmountedunmounted tiretire betweenbetween matchingmatching aluminum plates. VerticalVertical loadload waswas appliedapplied directlydirectly fromfrom thethe barebare rimrim toto thethe toptop aluminumaluminum plate.plate. Then,Then, thethe rimrim waswas pulledpulled laterally while while two two cap cap screw screw heads heads prevented prevented it from it from sliding sliding relative relative to the to upper the upper plate. plate. The Theapplied applied lateral lateral force force and andresulting resulting lateral lateral deflection deflection of the of rim the were rim were recorded recorded simultaneously. simultaneously. The Themeasured measured tread tread lateral lateral stiffness stiff nesswas then was thendivided divided by the by number the number of knobs of knobs present present between between the plates the platesto yield to an yield approximate an approximate individual individual knob stiffness. knob stiff ness. Appl. Sci. 2020, 10, 3156 16 of 22

Appl. Sci. 2020, 10, 3156 16 of 22 Appl. Sci. 2020, 10, 3156 16 of 22

Figure 13.13. SchematicSchematic illustrating non-destructive measurementmeasurement of lateral shear stistiffnessffness of tire treadtread Figure 13. Schematic illustrating non-destructive measurement of lateral shear stiffness of tire tread knobs viavia loadsloads appliedapplied toto unmountedunmounted tiretire squeezedsqueezed betweenbetween appropriatelyappropriately sizedsized aluminumaluminum plates.plates. knobs via loads applied to unmounted tire squeezed between appropriately sized aluminum plates. Finally, perper EquationEquation (8),(8), the tread shearshear stistiffnessffness waswas calculatedcalculated as springs in parallel by Finally, per Equation (8), the tread shear stiffness was calculated as springs in parallel by multiplying thethe individualindividual knobknob stistiffnessffness byby thethe numbernumber ofof knobsknobs evidentevident inin thethe tire’stire’s footprint.footprint. multiplying the individual knob stiffness by the number of knobs evident in the tire’s footprint.

Ktread ==nknobs ∗k knob ,, (8) (8) =∗∗ , (8) These respective stiffness values and the comparison of measured versus calculated total tire These respective stiffness stiffness values and the comparisoncomparison of measured versus calculated total tire stiffness are shown for the 29 × 2.3” knobby tire in Figure 14. stiffnessstiffness are shown for the 29 × 2.2.3”3” knobby tire in Figure 14.14. ×

Figure 14. Bar chart of knob, tread, bald carcass, measured, and calculated total tire stiffness for 29 × Figure 14. 14. BarBar chart chart of of knob, knob, tread, tread, bald bald carcass, carcass, measur measured,ed, and and calculated calculated total total tire stiffness tire stiff nessfor 29 for × 2.3” knobby tire comparing relative stiffness magnitudes at 25 psi (1.7 bar). 2.3”29 knobby2.3” knobby tire comparing tire comparing relative relative stiffness stiff nessmagnitudes magnitudes at 25 atpsi 25 (1.7 psi bar). (1.7 bar). × As can be seen, the approximate total tire stiffness matched well (within 8%) with the measured As can be seen, the approximate approximate total tire stiffness stiffness matched well (within 8%) with the measured lateral stiffness of the inflated knobby tire. Possible sources of variation included difficulty in lateral stiffness stiffness of thethe inflatedinflated knobbyknobby tire.tire. Possible Possible sources sources of of variation included difficulty difficulty in determining exactly how many knobs, or fractions thereof, were included in the contact patch. This determining exactly exactly how how many many knobs, knobs, or orfractions fractions thereof, thereof, were were included included in the in contact the contact patch. patch. This approach also assumed equal stiffness contribution from various knob typologies and ignored approachThis approach also alsoassumed assumed equal equal stiffness stiffness contributi contributionon from from various various knob knob typologies typologies and and ignored possible variation in load distribution across the patch. A similar investigation was not repeated for possible variationvariation inin loadload distribution distribution across across the the patch. patch. A A similar similar investigation investigation was was not not repeated repeated for thefor the file-tread pattern because of the impracticality of counting and measuring the stiffness of an the file-tread pattern because of the impracticality of counting and measuring the stiffness of an integral number of the file-tread knobs. integral number of the file-tread knobs. Appl. Sci. 2020, 10, 3156 17 of 22

file-tread pattern because of the impracticality of counting and measuring the stiffness of an integral number of the file-tread knobs. Appl. Sci. 2020, 10, 3156 17 of 22 3.3. Effect of Inflation Pressures and Rim Width 3.3. Effect of Inflation Pressures and Rim Width Tire inflation pressure is an important setup and tuning parameter for mountain bike applications. A happyTire compromiseinflation pressure between is traction,an important durability, setup protectionand tuning for parameter the rim, for handling, mountain and bike ride characteristicsapplications. A often happy depends compromise on rider between weight, traction, bike fitment, durability, terrain, protection even surface, for the rim, or soil handling, consistency, and andride of characteristics course, rider preference. often depends Potentially on rider for thisweight reason,, bike bicycle fitment, manufacturers, terrain, even unlike surface, automotive or soil orconsistency, motorcycle and companies, of course, rarely rider specify preference. a specific Potentia recommendedlly for this reason, operating bicycle inflation manufacturers, pressure forunlike each vehicleautomotive model or and motorcycle leave it to thecompanies, end-user rarely to experiment specify a within specific the recommended bounds typically operating designated inflation on the tirepressure sidewall. for Thus,each vehicle to better model explore and theleave potential it to the operatingend-user to range experiment of various within tires the mentioned bounds typically thus far indesignated this study, on additional the tire sidewall. measurements Thus, to were better taken explore at variousthe potential inflation operating pressures. range These of various results tires are mentioned thus far in this study, additional measurements were taken at various inflation pressures. reported in a format much like the previous sections. These results are reported in a format much like the previous sections. 3.3.1. Footprints 3.3.1. Footprints Figures 15 and 16 illustrate the effect of varying inflation pressure on the tire footprints for the 29 2.3”Figuresknobby 15 and tire on16 illustrate 25 mm rim. the Theeffect images of varying show inflation that as inflation pressure pressure on the tire increased footprints from for left the to 29× × 2.3” knobby tire on 25 mm rim. The images show that as inflation pressure increased from left to right, the contact patch length and width became smaller and thus contact patch area was reduced. right, the contact patch length and width became smaller and thus contact patch area was reduced. Interestingly, for the 29 2.3” knobby tire at 10 psi, in the leftmost plot, the tire deformed enough for Interestingly, for the 29× × 2.3” knobby tire at 10 psi, in the leftmost plot, the tire deformed enough for the shoulder knobs to contact the ground (even at zero camber), and thus the shape of the patch became the shoulder knobs to contact the ground (even at zero camber), and thus the shape of the patch significantly wider. As inflation pressure increased, fewer and fewer individual knobs remained in the became significantly wider. As inflation pressure increased, fewer and fewer individual knobs contact patch. remained in the contact patch.

Figure 15. Footprints for 29 × 2.3” knobby tire at inflation pressures of 10, 20, 30, 40, and 50 psi (0.7, Figure 15. Footprints for 29 2.3” knobby tire at inflation pressures of 10, 20, 30, 40, and 50 psi (0.7, 1.4, 1.4, 2.1, 2.8, and 3.4 bar) from× left to right on 25 mm inner width rim. 2.1, 2.8, and 3.4 bar) from left to right on 25 mm inner width rim.

FigureFigure 16 16 shows shows contact contact patchpatch characteristicscharacteristics as functions of of inflation inflation pressure. pressure. The The 29 29 × 2.3”2.3” × knobby,knobby, 29 29 ×2.3” 2.3” bald, bald, 29 29 × 2.5”2.5” file-tread,file-tread, andand 29 × 2.5”2.5” bald bald tires tires were were measured measured at at eight eight pressure pressure increments.× This gave added× resolution to various tr×ends. The other three knobby tires in 27.5 × 2.8”, increments. This gave added resolution to various trends. The other three knobby tires in 27.5 2.8”, 29 × 3”, and 26 × 4” are also shown, but with inflation pressure at just two levels, 10 psi (potential× use 29 3”, and 26 4” are also shown, but with inflation pressure at just two levels, 10 psi (potential use case× for lightweight× riders or usage on soft or loose terrain), and the “nominal” pressure considered case for lightweight riders or usage on soft or loose terrain), and the “nominal” pressure considered in in previous sections. Finally, two additional datapoints of the 29 × 2.3” knobby and 29 × 2.5” file-tread previous sections. Finally, two additional datapoints of the 29 2.3” knobby and 29 2.5” file-tread tire on slightly narrower, 22 mm internal rim widths are shown× to begin to explore those× effects. tire on slightly narrower, 22 mm internal rim widths are shown to begin to explore those effects. The upper left plot (Figure 16a) suggests the ranking of tires in terms of contact patch area varied somewhat with inflation pressure, but in general the contact patch area decreased in a seemingly nonlinear fashion with increasing inflation pressure. Here, the 26x4” fat bike tire remained near the largest, whereas the 29 × 2.3” knobby surpassed the plus tires between 10 and 20 psi. The contact patch area of the less-treaded tires in the form of the 29 × 2.5” file-tread on the 25 mm rim, 29 × 2.3” Appl. Sci. 2020, 10, 3156 18 of 22 bald, and 29 × 2.5” bald tire seemed slightly more sensitive in the lower pressure regime (from 10 to 15 psi), whereas as pressure was increased (beyond 20 psi), the curvature stabilized. The 29 × 2.3” knobby on 22 mm rim and 29 × 2.5” file-tread tire on 22 mm rims showed a small decrement in contact patch area with respect to their 25 mm rim variants. The upper right plot (Figure 16b) depicts how the tread void ratio within the contact patch area varied between tires and with inflation pressure. It is quite clear that the knobby tires had higher void ratios than the 29 × 2.5” file-tread on 25 mm rim at all but the lowest inflation pressure. The 26 × 4” fat bike tire and 29 × 2.3” knobby tire were both around 80% void ratio at low pressures, which for the 29 × 2.3” knobby tire on 25 mm rim fell to 74% at 50 psi. Seemingly, scaling the knob size up for the 27.5 × 2.8” and 29 × 3” plus tires caused a small reduction in their void ratio with respect to the 29 × 2.3” knobby tire on the 25 mm rim. The void ratio of the 29 × 2.5” bald tire, which contained only Appl.the negative Sci. 2020, 10 grooves,, 3156 reduced from 20% to 10% with increasing inflation pressure, whereas the 18nearly of 22 slick 29 × 2.3” bald tire hovered at a void ratio less than 5%.

(a) (b)

(c) (d)

FigureFigure 16. Tire 16. footprintTire footprint parameters parameters versus versus inflation inflatio pressuren pressure for various for various bicycle bicycle tires withtires (witha) contact a) patch area,contact (b) void patch ratio, area, (c) b) contact void ratio, patch c) length contact and patch (d) contact length patchand d) width. contact patch width.

TheThe upperlower lefttwo plot plots (Figure show 16 thea) suggestscontact patch the ranking length of(Figure tires in 16c) terms and of width contact (Figure patch area16d) variedversus somewhatinflation pressure. with inflation Here, pressure,the length but was in generally general the at least contact twice patch the areawidth decreased or more. in The a seemingly change in nonlinearlengths of fashion most ofwith the tires increasing shown inflationwere similar, pressure. whereas Here, the the 26 26x4”× 4” fat fat and bike 27.5 tire × 2.8” remained plus tires near were the largest,slightly whereas shorter thethan 29 others2.3” knobbyat the same surpassed inflatio then plus pressures. tires between The 29 10 × and2.3” 20 knobby psi. The tire contact retained patch a × areaslightly of the longer less-treaded contact patch tires in than the formthe 29 of × the2.5” 29 file-tread2.5” file-tread tire or their on the bald 25 counterparts mm rim, 29 at2.3” inflation bald, × × and 29 2.5” bald tire seemed slightly more sensitive in the lower pressure regime (from 10 to 15 psi), × whereas as pressure was increased (beyond 20 psi), the curvature stabilized. The 29 2.3” knobby on × 22 mm rim and 29 2.5” file-tread tire on 22 mm rims showed a small decrement in contact patch area × with respect to their 25 mm rim variants. The upper right plot (Figure 16b) depicts how the tread void ratio within the contact patch area varied between tires and with inflation pressure. It is quite clear that the knobby tires had higher void ratios than the 29 2.5” file-tread on 25 mm rim at all but the lowest inflation pressure. The 26 4” × × fat bike tire and 29 2.3” knobby tire were both around 80% void ratio at low pressures, which for × the 29 2.3” knobby tire on 25 mm rim fell to 74% at 50 psi. Seemingly, scaling the knob size up for × the 27.5 2.8” and 29 3” plus tires caused a small reduction in their void ratio with respect to the × × 29 2.3” knobby tire on the 25 mm rim. The void ratio of the 29 2.5” bald tire, which contained only × × Appl. Sci. 2020, 10, 3156 19 of 22

the negative grooves, reduced from 20% to 10% with increasing inflation pressure, whereas the nearly slick 29 2.3” bald tire hovered at a void ratio less than 5%. × The lower two plots show the contact patch length (Figure 16c) and width (Figure 16d) versus inflation pressure. Here, the length was generally at least twice the width or more. The change in lengths of most of the tires shown were similar, whereas the 26 4” fat and 27.5 2.8” plus tires were × × slightly shorter than others at the same inflation pressures. The 29 2.3” knobby tire retained a slightly Appl. Sci. 2020, 10, 3156 × 19 of 22 longer contact patch than the 29 2.5” file-tread tire or their bald counterparts at inflation pressures × pressuresabove 25 psiabove (1.7 25 bar). psi Finally, (1.7 bar). in termsFinally, of width,in term theres of width, was a clearthere grouping was a clear between grouping the widerbetween contact the widerpatch ofcontact the knobby patch of tires the and knobby the less-wide tires and patchthe less-wide of the file-tread patch of and the baldfile-tread variants. and bald variants.

3.3.2.3.3.2. Forces Forces and and Moments Moments FigureFigure 1717 showsshows the the stiffness stiffness coefficients coefficients for for slip and and camber camber forces forces as as well as self-aligning and and twistingtwisting torques torques versus versus inflation inflation pressure. pressure.

(a) (b)

(c) (d)

FigureFigure 17. 17. ForceForce and and moment moment fitted fitted stiffness stiffness coefficients coefficients versus versus inflation inflation pressure pressure for for various various bicycle bicycle tirestires with with (a ()a cornering) cornering stiffness, stiffness, (b) (camberb) camber stiffness, stiffness, (c) self-aligning (c) self-aligning stiffness sti ffandness (d and) twisting (d) twisting torque coefficients.torque coeffi cients.

TheThe upper upper left left plot plot (Figure (Figure 17a)17a) displays displays various various trends trends in in cornering cornering stiffness stiffness coefficient. coefficient. First, First, largelarge differences differences were were apparent apparent between between the the differen differentt tire tire sizes sizes bounded bounded on on the the upper upper end end by bythe the26 × 4” fat bike and 29 × 2.3” bald tires, with upper values in excess of 20 1/rad, and on the lower end by the 29 × 2.3” knobby tire, on either rim width, at around 10 1/rad. Aside from the 26 × 4” fat tire, the less-treaded tires had higher cornering stiffness above 20 psi (1.4 bar). Additionally, above this pressure, the cornering stiffness of the tires shown seemed to vary little with further increase in inflation pressure. Below this pressure, there were some nonlinear and disparate behaviors where cornering stiffness of the 29 × 2.5” file-tread increased rapidly with inflation pressure whereas that of the 29 × 2.3” knobby tire decreased gradually. Across the limited pressures measured, the 27.5 × 2.8” plus tire exhibited a different sensitivity (slope) than the other knobby tires. Appl. Sci. 2020, 10, 3156 20 of 22

26 4” fat bike and 29 2.3” bald tires, with upper values in excess of 20 1/rad, and on the lower × × end by the 29 2.3” knobby tire, on either rim width, at around 10 1/rad. Aside from the 26 4” fat × × tire, the less-treaded tires had higher cornering stiffness above 20 psi (1.4 bar). Additionally, above this pressure, the cornering stiffness of the tires shown seemed to vary little with further increase in inflation pressure. Below this pressure, there were some nonlinear and disparate behaviors where cornering stiffness of the 29 2.5” file-tread increased rapidly with inflation pressure whereas that of × the 29 2.3” knobby tire decreased gradually. Across the limited pressures measured, the 27.5 2.8” × × plus tire exhibited a different sensitivity (slope) than the other knobby tires. The plot of normalized camber stiffness, shown in the upper right (Figure 17b) showed similar overall groupings. The less-treaded variants exhibited higher camber stiffness coefficients, with a decreasing trend as inflation pressure increased, and the 29 2.3” bald tire topped the chart. Camber × stiffness of the knobby tires fell off more quickly from 10 to 25 psi, whereas the 29 2.3” knobby tire on × a 25 mm rim eventually achieved a slope similar to its less-treaded counterparts, albeit at a lower value (below 1.0 1/rad). Rim width had minimal effect. In the lower left plot (Figure 17c), the sign convention for the aligning torque should be kept in mind. Its value was negative because it acted to reduce the tire slip angle, and thus a larger stiffness magnitude resulted in a more negative value. Again, there was a grouping. Except for the 26 4” fat × bike tire at 10 psi, the less treaded tires exhibited more self-aligning stiffness, with increased inflation pressure, albeit with diminishing magnitude. The 29 2.3” and 27.5 2.8” knobby tires grouped fairly × × close together with the lowest magnitudes of self-aligning stiffness. It should be noted that although the units of normalized self-aligning stiffness were in meters, this value was different from the pneumatic trail. As described earlier, the pneumatic trail can be derived by dividing the tire self-aligning moment by the lateral force due to slip (at zero camber). Normalized twisting torque stiffness, shown in the lower right plot (Figure 17d), showed a fairly rapid decrease with increasing inflation pressure up to about 30 psi (2.1 bar) for the measured tires. On the basis of the higher twisting torque stiffness magnitudes of the 29 2.3” bald and 26 4” tires, × × one would expect them to generate the most steering torque at low inflation pressures. For the 26 4”, × this was likely related to the difference in peripheral velocity across such a wide patch, and for the 29 2.3” bald tire it may have been related to the relatively small toroid radius and steep shoulder × drop. Again, a minor change in rim width had minimal effect. Presumably, the steep slope of the 27.5 2.8” and 29 3” plus tires between 10 and 15 psi may have been related to the widely spaced × × shoulder knobs contacting at lower lean angles on the less inflated, softer, and more deformed tire.

3.3.3. Static Lateral Stiffness Section 3.2.4 described modelling the tire static lateral stiffness as multiple individual knobs acting as springs in parallel that constituted the tread stiffness. The tread stiffness then combined with the bald carcass stiffness as springs in a series to yield the total lateral stiffness of the treaded tire. Now, in Figure 18, that simplified model is applied across a range of inflation pressures, yielding results comparable to the measured value of the 29 2.3” knobby tire. The figure shows the individual knob × stiffness as derived from the tread shear measurement multiplied by the number of knobs in contact. On the basis of the footprints shown in Figure 15, “n”, the number of knobs in contact, plotted here on the secondary axis, decreased with inflation pressure. Hence, the product of those terms yielded the theoretical tread stiffness, which also decreased with inflation pressure. It is interesting to note that the individual knobs did not change in stiffness, but the fact that fewer of them were in contact reduced the effective tread stiffness. Meanwhile, the as-measured bald carcass stiffness was shown to increase with inflation pressure. Combining these two as springs in series resulted in a theoretical total tire lateral stiffness within 15% of the measured values across the range of inflation pressures. Appl. Sci. 2020, 10, 3156 21 of 22 Appl. Sci. 2020, 10, 3156 21 of 22

FigureFigure 18.18. Knob,Knob, tread,tread, baldbald carcass,carcass, measured,measured, andand calculatedcalculated totaltotal tiretire stistiffness,ffness, andand knobknob countcount forfor 2929 × 2.3” knobby tire comparingcomparing stistiffnessffness magnitudesmagnitudes acrossacross inflationinflation pressures.pressures. × 4. ConclusionsAs before, possible sources of the difference between calculated and measured total stiffness included difficulty in determining exactly how many knobs, or fractions thereof, were included in Characterization of four modern mountain bike tire sizes was carried out on appropriate rim the contact patch and, for low inflation pressures, the participation of shoulder knobs, whose shear widths at what were considered realistic nominal inflation pressures. Tire cross sections were stiffness was not measured. compared, and tire radii and width values reported. Tire footprints at zero camber angle and realistic 4.normal Conclusions load were fit as ellipses and their area, geometry, void ratios, and tread patterns were discussed. Force and moment measurements were conducted using the measuring device at the Characterization of four modern mountain bike tire sizes was carried out on appropriate rim University of Wisconsin–Milwaukee. A combination of Pacejka’s Motorcycle Magic Tire Model and widths at what were considered realistic nominal inflation pressures. Tire cross sections were compared, FastBike twisting torque polynomial were fitted to the data to yield parametric tire model coefficients. and tire radii and width values reported. Tire footprints at zero camber angle and realistic normal These fitted curves were then used to compare lateral force due to slip and camber, as well as self- load were fit as ellipses and their area, geometry, void ratios, and tread patterns were discussed. aligning moment and twisting torque. Good resolution of the linear portion of the respective stiffness Force and moment measurements were conducted using the measuring device at the University of curves were obtained, whereas tire versus treadmill width limited identification of curvature and Wisconsin–Milwaukee. A combination of Pacejka’s Motorcycle Magic Tire Model and FastBike twisting peak friction values to some extent. Static lateral and radial carcass stiffnesses were shown. The torque polynomial were fitted to the data to yield parametric tire model coefficients. These fitted curves compilation of these results suggests appreciable differences in tire performance among 29 × 2.3”, were then used to compare lateral force due to slip and camber, as well as self-aligning moment and 27.5 × 2.8”, 29 × 3”, and 26 × 4” knobby tires. twisting torque. Good resolution of the linear portion of the respective stiffness curves were obtained, Additional characterization was carried out to quantify the effects of tire tread knobs. A 29 × 2.3” whereas tire versus treadmill width limited identification of curvature and peak friction values to knobby tire was compared to a file-tread tire of similar size. The treads were then removed from both some extent. Static lateral and radial carcass stiffnesses were shown. The compilation of these results tires, and characterization of the resultant bald tires further informed this comparison. It was shown suggests appreciable differences in tire performance among 29 2.3”, 27.5 2.8”, 29 3”, and 26 4” that file-tread and a tread pattern with negative grooves had appreciable× void× ratios and× performance× knobby tires. trade-offs that were different from a truly bald, slick tire. It was also shown that for the 29 × 2.3” Additional characterization was carried out to quantify the effects of tire tread knobs. A 29 2.3” knobby mountain bike tire studied, the combination of tire tread shear stiffness, obtained× by knobby tire was compared to a file-tread tire of similar size. The treads were then removed from both combining individual knobs as springs in parallel, with the inflated bald tire carcass as springs in tires, and characterization of the resultant bald tires further informed this comparison. It was shown series, yielded a lateral stiffness value similar to that measured for the total treaded tire. that file-tread and a tread pattern with negative grooves had appreciable void ratios and performance Finally, the aforementioned footprint, force and moment, and static carcass lateral stiffness trade-offs that were different from a truly bald, slick tire. It was also shown that for the 29 2.3” parameters for each tire were compared across a range of inflation pressures. Results suggest× some knobby mountain bike tire studied, the combination of tire tread shear stiffness, obtained by combining differences between knobby and less-treaded tires, especially at higher inflation pressures. Inflation individual knobs as springs in parallel, with the inflated bald tire carcass as springs in series, yielded a pressure influenced the number of knobs interacting with ground as the toroidal tires deformed. lateral stiffness value similar to that measured for the total treaded tire. Incorporating this observation into the approximation of tire lateral stiffness as a combination of tread Finally, the aforementioned footprint, force and moment, and static carcass lateral stiffness shear stiffness and carcass lateral stiffness as springs in series seemed promising. A brief comparison parameters for each tire were compared across a range of inflation pressures. Results suggest some of a 25 mm and 22 mm rim suggested small relative differences for the two tires measured. differences between knobby and less-treaded tires, especially at higher inflation pressures. Inflation Ideally, this work supplements ongoing progress in bicycle modelling and simulation [12]. pressure influenced the number of knobs interacting with ground as the toroidal tires deformed. Parametrization of nonlinear tire properties, as shown here, may aide higher fidelity bicycle and even Incorporating this observation into the approximation of tire lateral stiffness as a combination of tread mountain bike handling and stability simulations. A further study looking at alternate tire sizes, specifications, and tread patterns; isolating individual knob typologies; and implementing a wider Appl. Sci. 2020, 10, 3156 22 of 22 shear stiffness and carcass lateral stiffness as springs in series seemed promising. A brief comparison of a 25 mm and 22 mm rim suggested small relative differences for the two tires measured. Ideally, this work supplements ongoing progress in bicycle modelling and simulation [12]. Parametrization of nonlinear tire properties, as shown here, may aide higher fidelity bicycle and even mountain bike handling and stability simulations. A further study looking at alternate tire sizes, specifications, and tread patterns; isolating individual knob typologies; and implementing a wider treadmill would be interesting. Applying similarity methods to link tire performance to width and radius may warrant additional study, whereas a design of experiments (DOE) approach focused on the effects of specific construction parameters would benefit from bicycle tire manufacturer support in building relevant test samples.

Author Contributions: Conceptualization, J.S. and A.D.; methodology, A.D.; software, A.D. and J.S.; validation, A.D. and J.S.; formal analysis, J.S. and A.D.; investigation, A.D. and J.S.; resources, A.D.; data curation, A.D. and J.S.; writing—original draft preparation, J.S. and A.D.; writing—review and editing, A.D. and J.S.; visualization, J.S.; supervision, J.S.; project administration, A.D.; funding acquisition, J.S. All authors have read and agreed to the published version of the manuscript. Funding: This research received no external funding. Conflicts of Interest: The authors declare no conflict of interest.

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