Guide for Pavement Friction (2009)

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E-1 APPENDIX E PRIMER ON FRICTION

E-1 INTRODUCTION This primer is based in part on a 1999 report written for the Joint Winter Runway Friction Measurement Program and the American Society for Testing and Materials (ASTM) (Andresen and Wambold). It is intended to provide background material for users of braking slip measurement devices, with emphasis on topics related to comparison and harmonization of friction measurement devices. It describes different aspects of measuring braking slip friction on traveled surfaces. In practice, all types of surfaces and conditions are encompassed, ranging from bare and dry to pavements covered with precipitation deposits, and thus a year-round context is provided. The mechanics of various combinations of tire–surface interaction mechanisms are discussed. A step-by-step, parallel case presentation of a force-measuring friction device and a torque-measuring friction device highlights the difficulties of obtaining mechanical error-free measurements of braking slip friction. Models for the interaction between a braked tire and a surface are presented and discussed, as are several approaches to harmonization of friction measuring devices. The International Friction Index (IFI), as proposed by the World Road Association (PIARC), is presented and discussed. Who is This Primer For? Although the generated friction phenomenon between a braked wheel and a traveled surface may be covered in many different textbooks of science and engineering, no single source has been found that is dedicated to the measurement of braking slip friction. There are hundreds of research papers that report on various aspects of braking slip friction. This primer does not purport to reflect an overview or summary of findings from a literature study. Rather, it is a collection of topics that were visited by the authors during planning of field tests, analysis of collected data and design work for a harmonized unit of friction measure and the standards development process within ASTM Committee E17 on Vehicle- Pavement Systems. The primer covers elementary mechanics, dynamic influences on friction by winter contaminants, physical modeling of friction, elements of applied statistics, variability of friction measures and standard friction measures. Since the treatment of these topics seeks to establish sound ways of comparing and harmonizing friction measurements, some aspects may require further investigation or careful evaluation before they are fully accepted in the field of tire–surface friction measurement. This primer is intended to serve as a guide or discussion text for researchers, tire–surface measurement method designers, equipment manufacturers, and operators in the field of measuring braking friction by public service regulators, highway engineers, maintenance personnel, and other users of road and highway friction information. Andresen, A. and J.C. Wambold. 1999. “Friction Fundamentals, Concepts, and Methodology,” Report TP 13837E, Transportation Development Centre, Transport Canada, Ottawa, Ontario, Canada.

E-2 The Friction Measurement Devices A number of different types of devices have been invented and deployed at different highway agencies to provide information about the road surface frictional characteristics. Few devices have been designed specifically for predicting ground vehicle braking performance. These devices have to meet demands for ease of use, low cost of purchase and maintenance, consistency of measured results, and reliability of operation. Devices that measure acceleration during a change of velocity or that measure force for a continuous braked wheel have become very popular. In this primer, the focus is on the common characteristics of devices rather than on the individual device types used for friction measurement. The goal is to establish a basis for harmonization of their outputs. The friction measurement devices used in road and highway friction measurements can be grouped into five families: • Locked-wheel testers that have a fully braked wheel measuring short segments of the road periodically. • Side-force friction measurement devices that have a test wheel, mounted in line with the wheel track and angled to the direction of travel, under a known load. These devices can be related theoretically to the fixed-slip measurement devices. • Fixed-slip testers that have a fixed and continuous level of applied braking on the measuring wheel. • Variable-slip testers that have a variable controlled level of braking, usually with a governing time function that is repeated in continuous cycles. • Decelerometer testers, where the brakes of the host vehicle are applied sufficiently hard to lock the wheels and retard the vehicle for a short distance and time. The vehicle is accelerated to the same initial speed before another deceleration is initiated. The braking slip testers (both fixed and variable) are typically outfitted with strain gauges to measure the following: • One force parallel with the surface, using the static weight as the normal load; • Two forces, one parallel with the surface and one normal to the surface; • One torque measurement of the wheel braking moment using the static weight as the normal load; or • Combination of force and torque measurements. Focus on Braking Slip Friction Measurements of friction, as reported by friction testers, are really aggregated measurements of different forces induced by motion that are present in variable quantities for different pairs of braked wheels and surfaces. The purpose of braking a wheel is to make controlled use of what may be called the braking slip friction. Other forces induced by the motion are not controllable and constitute unwanted influences on the braking slip friction. The rolling resistance stems from the mechanics of the rolling tire.

E-3 Several other resistive forces can be important to the stopping of a vehicle. Aerodynamic drag and impingement drag are examples of non-frictional stopping forces. Discussion regarding the stopping of a vehicle is outside the scope of this primer; the reader is referred to textbooks on vehicle dynamics that cover stopping a vehicle. THE NATURE OF BRAKING SLIP FORCES Main Mechanisms of Braking Slip Friction Although the mechanisms of braking slip friction is not fully understood, the process is regarded by many experts as a composition of three main elements: • Adhesion. • Hysteresis. • Shear (wear, tear). Figure E-1 depicts these mechanisms in the tire–surface interface. The shear is indicated for a non-rigid surface material only. Adhesion Hysteresis Slip speed Surface material Tire tread Shear Figure E-1. An exploded view of a tire–surface interface. The braking slip force, FB, can be viewed as a sum of three terms: shearhysteresisadhesionB FFFF ++= Eq. E-1 Surface texture influences all three mechanisms. The adhesion force is proportional to the real area of adhesion between tire and surface asperities. The hysteresis force is generated within the deflecting and visco-elastic tire tread material and is a function of speed. The shear force is proportional to the area of shear developed. Generally, adhesion is related to micro-texture whereas hysteresis is mainly related to macro-texture. For wet pavements, adhesion drops off with increased speed while hysteresis increases with speed, so that above 56 mi/hr (90 km/hr), the macro-texture has been found to account for over 90 percent of the friction. In the case of winter friction on snow and ice, the shear strength of the contaminant is the limiting factor.

E-4 Figure E-2 depicts typical compositions of the braking slip friction mechanisms for two different surfaces interacting with the same tire. The pie chart on the left depicts a rigid surface, such as a dry, bare pavement. The pie chart in the middle depicts a wet pavement. The pie chart on the right depicts a non-rigid surface material. For a tire tread in contact with a rigid surface, the shear force is usually regarded as small. Adhesion and hysteresis make up 80 to 90 percent of the braking slip force. Pieces of tire tread are torn off when interfacing with a rigid surface. The tire is therefore called the sacrificial part of the braking slip friction process. Adhesion Hysteresis Shear Adhesi Hysteresis Shear Adhesion Hysteresis Shear AdhesionHysteresis Shear Figure E-2. Three theoretical sample compositions of major influences on braking slip. A significant shear force contribution implies that the sacrificial component is being sheared. In other words, the shear force is proportional to the product of the ultimate shear stress of the surface material and the real area of shearing contact. Because of the markedly different compositions of braking slip mechanisms for rigid versus non-rigid surfaces, a question is raised whether the braking slip process can be considered sufficiently uniform for different compositions to be included in the same comparison of friction testers. Intuitively, the nearly same compositions of braking slip mechanisms would produce the best correlations. Thus, comparison of devices on compacted or rolled snow would differ from comparisons on pavement. Simple Friction Models Amontons Friction Model The simplest friction model for two objects in contact and undergoing opposing movement is the familiar Amontons1 friction model. It states that the pulling force required to sustain an opposing motion of a pair of interfacing objects is directly proportional to the perpendicular contact force. This pulling force is called the friction force and is independent of the apparent contact area. The factor of proportion has been named the 1 Guillaume Amontons, French physicist, 1699.

E-5 coefficient of friction, μ. In figure E-3, the perpendicular contact force is the weight of the block, FW. The Amontons equation is as follows: WFF ⋅= μ Eq. E-2 The friction is a measure of the resistive interaction of the interfacing objects. The friction is a characteristic of the two objects. The Amontons equation works best for solid objects. FW F fw f Figure E-3. Pairs of objects of same material of different size and weight having the same and constant coefficient of friction. This friction model is commonly used to estimate the force required to sustain the opposing motion when the perpendicular contact force and the friction coefficient for the interfacing materials are known. If the Amontons model holds true equally well both for the friction measurement device and the vehicle–tire interaction with the road surface, a friction coefficient acquired with the ground friction measurement device could be applied to a vehicle tire and suspension (see figure E-4). In the interaction between a pneumatic tire and a surface, dependencies on many parameters are encountered for the friction coefficient. This makes the Amontons friction model invalid for application with pneumatic tires. It is evidenced by the fact that different types of friction measurement devices report different values of the friction coefficient when measuring the same surface. In essence, this is the reason why it is necessary to transform friction values to a common unit of measure. It must be acknowledged that each type of device equipped with a tire has its own proprietary set of reported numbers expressing friction. There are several reasons for this diversity. The flexible tire object manufactured from visco-elastic materials is a cause of non-linearity. The irregularity of the surface, called texture, is another major factor. Different tire–surface pairs exhibit different non-linearity characteristics of friction. Wet pavement with low texture content against a bald tire tread, for example, will have a pronounced reduction in the coefficient of friction as travel speed increases.

E-6 Figure E-4. Two wheels of different size and type on the same surface. Friction is a phenomenon of surfaces in contact under opposing motion. The relative motion is called slip speed. Slip Speed and Slip Ratio The difference in tangential speed for a point on the tire circumference in the contact area when it is free-rolling versus braked at a constant travel speed of the wheel axis is called slip speed. The tangential speed for a free-rolling tire is equal to the travel speed. When the tire is braked, its tangential speed is less than the travel speed, as the travel speed is kept constant. When V is the travel speed and VB is the tangential speed of the tire when braked, the slip speed, S, is V - VB. The tangential speed is the rotational speed, ω, multiplied by the deflected tire radius, r. ω ωωω BBB rrVVS −=⋅−⋅=−= 1 Eq. E-3 By measuring the rotational speeds of the tire in free-rolling mode, ω, and braked mode, ωB, the slip speed can be calculated with the above equation. The ratio of the slip speed to the travel speed is called a slip ratio, λ. It can be expressed as follows: V V V VV V S BB −=−== 1λ Eq. E-4

E-7 Friction as Function of Travel Speed and Slip Speed Figure E-5 illustrates how braking slip friction can vary with travel speed and degree of braking, in terms of slip ratio. This figure suggests that a simplified, universal friction model for tire–surface object pairs can be expressed with a speed variable and a degree of braking called slip speed. With reference to figure E-4, where FW and fw are the weights of the vehicles on the wheels, the resistive forces for each tire-device configuration are ( ) ( ) WL FV,SV,SF ⋅≈ μ Eq. E-5 and ( ) ( ) wS fV,SV,Sf ⋅≈ μ Eq. E-6 1.2 1.0 0.8 0.6 0.4 0.2 20 40 60 80 1000 35 km/h 65 km/h 85 km/h B ra ki ng s lip fr ic tio n Percent slip Figure E-5. A case of braking slip friction with automotive tires on a dry surface (Clark, 1981). Since different friction measuring tires measure different friction values because of differences in contact area, rubber compound, and other parameters, then ( ) ( )V,SV,S SL μμ ≠ Eq. E-7 Therefore, ( ) ( ) WS FV,SV,SF ⋅≠ μ Eq. E-8 Clark, S.K. 1981. “Mechanics of Pneumatic Tires,” U.S. Department of Transportation, National Highway Traffic Safety Administration, Washington, D.C.

E-8 There are circumstances in which the friction has negligible influences of traveling speed and degree of braking, but for a universal friction model those circumstances are special cases. To circumvent this, a frequent tactic is to fix the measuring speed and slip speed and compare the device-tire configurations at those speeds. Vehicle Braking Friction When braking a vehicle to stop from low speeds on a level surface, the braking slip friction force generated in the tire–surface interaction equals a decelerating force acting on the vehicle mass: ondeceleratibraking FF = Eq. E-9 The applied braking force is equal to the deceleration force of the vehicle body mass according to Newton’s law: a g FF WW ⋅=⋅μ Eq. E-10 where g is the gravitational constant and a is the deceleration. Simplifying the expression, the friction coefficient is as follows: g a=μ Eq. E-11 This is a popular relationship used in determining the average friction coefficient (over the speed range) by measuring the deceleration of the vehicle. It is also frequently used in rough estimates of the average braking performance of vehicles in terms of deceleration on a surface, assuming the friction coefficient is valid for the vehicle-surface pair. At higher speeds, or when better accuracy is required, the braking equation can include other resistive terms such as aerodynamic resistance, longitudinal slope of the surface, displacement drag from liquid, fluid or plastic materials, impingement drag on the vehicle body from loose surface material, hydroplaning effects, brake efficiencies, weigh-in-motion and other parameters. Since these effects are not braking slip friction by nature, they must be assessed and used to correct a measured deceleration value to determine a braking slip friction coefficient. Or, when modeling the stopping of a vehicle, the effects of non-friction influences must be properly included. MECHANICS OF TIRE–SURFACE INTERACTIONS To understand the braking slip friction processes on a macro-scale, it is helpful to look at the mechanics of the interaction between a braked wheel with a pneumatic tire and different surface types and conditions. The material covered here is general. An actual friction measuring device design will have a unique geometry and a unique suspension that will require its own unique elaboration of mechanics. A distinction is made between force- measuring devices and torque-measuring devices. The different features of these two

E-9 groups of devices are highlighted. Intermittent or spot friction measuring devices are not fully addressed in this primer. In this primer, torque refers to measured moments transmitted by an axle. Applied torque on an axle to produce braking is referred to as an applied moment. Mechanics of a Wheel in a Constant and Continuous Measuring Mode Continuously measuring friction measurement devices operate at a constant travel speed. Furthermore, fixed-slip devices have no angular acceleration of the measuring wheel. Therefore, fixed-slip continuous friction measurement devices may be studied in steady- state equilibrium. The next few sections treat individual aspects in a cumulative manner, starting with a free- rolling tire, then adding drag, planing from a fluid contaminant and, finally, brake actuation. Rolling Resistance Even when free-rolling on a hard, non-contaminated surface, there is a resistive force to the tire movement. This is due to the natural and characteristic deflection of a pneumatic tire when rolling. Figure E-6 shows the forces acting on a wheel and tire. The host vehicle pulls the tribometer at a constant speed with force FX. The normal load on the measuring wheel is FW. r a FG Fwω FM Free body diagram, steady stateDirection of travel Pavement r a ω FR Fw Normal pressure distribution when rolling Fx Figure E-6. Rolling resistance force with a free-rolling tire at constant speed.

E-10 A small longitudinal tire slip force in the footprint supports the deflection work. As a result, the normal pressure distribution becomes uneven, such that the resultant normal force (center of pressure) from the ground, FG, is leading the vertical through the wheel center, and thereby creates a balancing resistive moment. The distance a by which the resultant force is leading the wheel axle is increasing with accelerating travel speed. The rolling resistance moment, FG·a, must be opposed with a moment, FR·r, applied about the wheel axis, if the wheel is to maintain a constant rotation and travel speed typical of continuous friction measurement devices. The wheel in figure E-6 can only produce this opposing moment by tire slip in the contact area when wheel-bearing resistance is disregarded. The surface is reacting to the slip with the force FR. If the surface is incapable of sustaining this slip, the wheel will not rotate. It will instead slide in its load-deflated state. This rarely happens, since the attainable friction force in all practical cases is greater than FR. Summation of the moments about the wheel axis yields 0=⋅−⋅ rFaF RG Eq. E-12 There is no torque transmitted over the wheel axle to other shafts or axles. A torque- measuring friction device is designed to measure the axle torque and therefore would measure zero. Solving for FR, GR Fr aF ⋅= Eq. E-13 This equation is a definition of tire rolling resistance. The resistive slip force, FR, is equal to the ground reaction force, FG, multiplied by a ratio of geometric parameters, a/r. In this scenario FG = FW , and therefore by substitution into equation E-13, the tire rolling resistance for a free-rolling case can be written as: WR Fr aF ⋅= Eq. E-14 From summation of horizontal forces in a steady-state equilibrium, 0=− MR FF Eq. E-15 Or rewritten, RM FF = Eq. E-16 A force-friction measuring device can measure the rolling resistance force if the design allows the applied brake moment to be uncoupled.

E-11 Since the nature of the tire rolling resistance involves slip in the tire–surface contact area, a friction coefficient can be defined as follows: r a F F F F W Wr a W R R = ⋅==μ Eq. E-17 The tire rolling resistance is geometrically defined. Both a and r may vary with tire design, tire load, speed, degree of braking, influence of contamination, etc. For dry, rigid horizontal surfaces, the rolling resistance is typically observed to be in the range of 0.5 to 3 percent of the carried weight. The tire rolling resistance is a tire property and is called tire rolling resistance for clarity to differentiate it from other forms of resistance to rolling, stemming from influences of contaminants as described in later sections. The tire rolling resistance is associated with the presence and location of the ground reaction force, FG, in the rigid surface contact region with a tire. Applied Braking Force To measure braking slip friction, a friction measuring device must apply a braking moment. A scenario with braking is depicted in figure E-7. A constant applied braking slip force, FB, works opposite to the rotation of the wheel. The applied brake moment, MB, is the product of the applied force and the radius of the sprocket wheel. The tire rolling resistance force couple (FR and FM) is always present when the wheel is rotating. r a FG Fwω FM Free body diagram, steady stateDirection of travel Pavement r a ω FR Fw Normal pressure distribution when rolling Fx FB MB FA Sprocket wheel brake transmission Figure E-7. Forces and moments of a constant braked wheel on a clean and dry rigid surface.

E-12 A brake moment causes the wheel rotation to slow down and creates a slip resistive force, FB, in the tire–surface contact area. An increased pulling force, FX, is required to uphold the tribometer at a constant speed of travel. Summing the moments about the wheel axis in equilibrium at steady state, 0=⋅+⋅−⋅− aFrFrFM GRBB Eq. E-18 A torque-measuring friction device will, by design, measure the reaction of the applied brake moment, called the measured torque, TM, that is equal to the applied brake moment, MB. Solving the above equation for MB or TM, aFrFrFMT GRBBM ⋅−⋅+⋅== Eq. E-19 If it can be assumed that the tire deformation during braking has the same basic relationship for tire rolling resistance as for the free-rolling case, and FG acts in the vertical plane only, then FR⋅r equals FG⋅a and the measured torque is as follows: rFT BM ⋅= Eq. E-20 A torque-measuring friction device does not measure tire-rolling resistance. The braking slip force is equal to the measured torque divided by the deflected radius. By summation of horizontal forces at steady state 0=−− RBM FFF Eq. E-21 Solving for FM, BRM FFF += Eq. E-22 A force-friction measuring device measures braking slip and tire-rolling resistance. When the objective is to measure braking slip, the tire-rolling resistance is an error term. At this point it is instructive to note a simple way to determine tire-rolling resistance by designing and building a friction tester to measure both torque and horizontal force. Solving for FR in the above equation and substituting for FB using equation E-20, r TFF MMR −= Eq. E-23 So far, the forces and moments due to the rolling resistance force and applied braking moment have been discussed. This scenario is valid for friction measurements of clean, dry and rigid surfaces. Next, drag forces due to fluid contaminant displacement will be studied when a measuring wheel is kept free rolling at a constant speed. This is useful for investigations of displacement drag parameters.

E-13 Friction Forces from Contaminant Dynamic Planing2 Planing occurs when the fluid3 contaminant material is trapped under the rolling tire in sufficient quantities at a high enough traveling speed to detach some or the entire tire tread from the base surface. Some, or all, of the tire rides on the trapped fluid contaminant, which acts like a lubricant. The fluid contaminant gets trapped because there is insufficient time for the fluid to flow out of the footprint area. Also, the surface and tire tread may not have sufficient grooves or voids to allow the fluid to fill into these spaces, and thus escape readily from the tire footprint area. As the trapped fluid enters the leading edge of the contact area between tire and surface, it gives rise to a fluid lift force acting to separate the tire from the base surface. When the fluid penetration covers all of the contact area with the ground, the tire–surface friction becomes approximately zero. The travel speed in this instance is called the critical hydroplaning speed when the fluid is water. A scenario dealing with the mechanics of friction tester tires with fluid planing is depicted in figure E-8. This is a free-rolling tire with no brake applied. A major difference from earlier scenarios is the divided reaction force from the ground. There are two forces, FG and FL, that carry the normal load, FW. FG is the ground reaction force from the base surface still in contact with the tire. FL is a resultant dynamic fluid lift force from the area of interspersed fluid. The line of attack for the ground reaction force, FG, is shifted back in the contact length, distance a from the vertical line through the wheel axis. As a result of this shift in location of FG, the tire rolling resistance force, FR, acts counterclockwise in figure E-8. The fluid lift force has a line of attack that is a distance, b, from the vertical line through the wheel axis. The fluid lift force has horizontal ground reaction force, FLG, acting in the tire–surface contact area. The sum of FR and FLG constitutes a resultant rolling resistance force. The fluid lift force, FL, sustains no shear forces in its contact area with the tire and, therefore, no slip to support tire-rolling resistance in this area. Assuming that it acts in the center of the interspersed fluid contact area, FL always acts ahead of the vertical through the wheel axis until full planing has occurred. At full planning, it acts vertically through the wheel axis. As the planing progresses, FG reduces to zero at full planing. The line of attack for the resultant normal reaction force is therefore always ahead of the wheel axis position. In that position, it resists the rotation of the wheel in the same manner as tire-rolling resistance when there is no fluid present. 2 Called hydroplaning when the fluid is water. 3 It is debatable whether to consider and call the different loose winter contaminant material fluids. Here it is used to associate with the established engineering for fluids.

E-14 r Direction of travel ω r ω a b FM Free body diagram, steady state Base surface (pavement, ice, compacted snow) Simplified normal pressure distribution when rolling Fluid cover (water, slush, loose snow) Fx FDFR FG FW FL FDG FW FLG Figure E-8. Free-rolling friction measuring device wheel with fluid lift and drag. The reaction force arising from the remaining ground contact, FG, has a line of attack before and after the roll axis position, depending on the degree of planning. The sum of surface reaction forces is equal to the static weight carried by the wheel: LGW FFF += Eq. E-24 A torque-friction measuring device measures zero, as all terms in a summation of moments about the wheel axis reduce to zero. The fluid lift force, FL, has a reaction force, FLG, in the contact surface between tire and ground. By taking the moment about the wheel axis, rFbF LGL ⋅=⋅ Eq. E-25 Rearranging the above equation, the horizontal fluid lift reaction force is as follows: LLG Fr bF ⋅= Eq. E-26 Summation of horizontal forces in equilibrium at steady-state yields: 0=−−++ LGDDGRM FFFFF Eq. E-27 With a fluid lift and drag acting on the tire, the horizontally measured force is determined as: RLGDGDM FFFFF −+−= Eq. E-28

E-15 Substituting for FDG and for FLG using equation E-26, and simplifying, RLDM FFr bF r tF −⋅+⋅⋅= 2 Eq. E-29 Since GR Fr aF ⋅= and LWG FFF −= , then ( )LWR FFr aF −⋅= Eq. E-30 Substituting for FR using equation E-30 in equation E-29, and simplifying, WLDM Fr aF r baF r tF ⋅−⋅++⋅⋅= 2 Eq. E-31 A force-friction measuring device with a de-coupled brake measures effects of displacement drag, fluid lift and planing. The Nature of the Fluid Lift Force Using Petroff’s equation (given in Goodenow et al., 1968) for bearing lubrication, the fluid dynamic lift force can be expressed as: VArkF LLL ⋅⋅⋅⋅= ρ Eq. E-32 where kL is the fluid dynamic lift coefficient, ρ is the fluid mass density, AL is gross tire-fluid contact area and V is the travel speed. The fluid dynamic lift coefficient depends on fluid viscosity and has a unit 1/time. The propagation of planing is different for different tire designs; therefore, there is no fixed general relationship between the offset distances a and b from the vertical through the wheel axis. See the section titled “Fluid Planing with Different Tire Designs” for a discussion of planing contact area for different tires. Horne and Dreher (1963) discuss two effects of water on tire-pavement interaction. One effect is hydroplaning, where inertia of the wheel and density properties of the fluid Goodenow, G., T. Kolhoff, and F. Smithson. 1968. “Tire–Road Friction Measuring Systems—A Second Generation,” Society of Automotive Engineers (SAE) Paper No. 680137, SAE. Horne, W.B. and R.C. Dreher. 1963. “Phenomena of Pneumatic Tire Hydroplaning,” NASA TN D- 2056, National Aeronautical and Space Administration (NASA).

E-16 predominate. The other is thin film lubrication, where viscous properties of the fluid predominate. The Moving Position of the Fluid Lift Force A linear relationship between speed and the propagation of the planing front under high- pressure tires can be assumed. At full planning, the lift propagation length l = L. The speed at full planing is called the critical planing speed, VC. The ratio of the propagation length to the full length is set equal to the ratio of measuring speed to critical planing speed. This can be expressed as follows: CV V L l = Eq. E-33 or, solving for l, L V Vl C ⋅= Eq. E-34 To build a mathematical model of the fluid lift force, it can be assumed that the lift force is proportional to the separation area (length l, width w), the speed and the density of the fluid. It is also proportional to the curvature of the lift area (i.e., FX or FM the radius of the tire). Thus, VrlwkF LL ⋅⋅⋅⋅⋅= ρ Eq. E-35 Using equation E-34 to substitute for l and setting ρ⋅V=1, since density and speed effects are already included in VC: C LL V VLrwkF ⋅⋅⋅⋅= Eq. E-36 The group w⋅r⋅L represents geometric tire properties and can therefore be included in a new tire coefficient, kPL, such that: C PLL V VkF ⋅= Eq. E-37 This equation is a model equation to study the fluid lift as a dependent variable of travel speed and a set of constant parameters for a given tire configuration. Fluid Lift Effects on the Tire–Surface Friction When Free Rolling The fluid lift phenomenon reduces the contact area for supporting the tire–surface slip resistive forces. In a free-rolling mode the only resistive force is due to rolling resistance

E-17 when disregarding fluid displacement drag. The tire-rolling resistance coefficient of friction is as follows: G R R F F=μ Eq. E-38 With no fluid lift, drag or brake, this equation represents the tire-rolling resistance slip friction coefficient on a clean surface. It is then a maximum attainable value, μRlim. The tire-rolling resistance slip friction force is: GRR FF ⋅= limμ Eq. E-39 But in this scenario, FG is equal to FW – FL , and therefore: ( )LWlimRR FFF −⋅= μ Eq. E-40 Substituting FL with equation E-37 gives: ⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ ⋅−⋅= C PLWRR V VkFF limμ Eq. E-41 When considering a tire configuration with a constant normal load, the braking slip friction force equation can be rearranged and a factor, kX, introduced, defined as: W PL X F kk = Eq. E-42 Then, the friction force equation becomes the following: ⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ ⋅−⋅⋅= C XWlimRR V VkFF 1μ Eq. E-43 At the boundary condition of full planing where V = VC, kX must be equal to one for FR to be zero. Thus, the general braking slip friction force equation is: ⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ −⋅⋅= C WlimRR V VFF 1μ Eq. E-44 or, expressed in terms of a fluid planing ratio, kP, for a free-rolling wheel, C P V Vk = Eq. E-45 Therefore, by substitution,

E-18 ( )PWlimRR kFF −⋅⋅= 1μ Eq. E-46 Thus, the fluid lift or planing effects on the friction characteristics amount to a reduction of the slip friction force equal to a fraction of the maximum attainable friction force value for the surface that is proportional to the planing ratio. For a force-measuring friction device, the measured friction, FM, is equal to FR when disregarding fluid displacement drag effects. Figure E-9 shows that the braking slip friction diminishes as the partial planing progresses, and that it is proportional to the speed and inversely proportional to the critical planing speed for the tire–surface combination. The VC parameter is a constant parameter for the tire–surface combination. V/VC 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 F R (V /V C ) ( N ) 0 10 20 30 40 Scenario: μRlim=0.025 FW=1500 N Figure E-9. A case of diminishing tire-rolling resistance force FR as the planing propagates with increased speed. Dynamic Contaminant Planing Propagation The effect of dynamic contaminant planing is the loss of contact area for any braking slip friction to be generated. The tire configuration has a significant influence on the system of forces generated. To study this influence, two tire types will be selected that differ in the way they deform in the contact zone when planing. The position of the resultant fluid lift force will shift with travel speed and other parameters. The tire is generally stronger along the sidewall than in the center of the contact patch. The weaker center area yields to the impingement forces of the fluid and allows it to penetrate under the center of the tire contact area. The penetration of fluid lifts the tire from the leading edge and separates the tire from the ground with the interspersed fluid. The sidewalls carry normal load to press the area along the sidewall to the ground.

E-19 Effectively, the fluid gets trapped in the center where the tire is weaker. If it is present in sufficient amounts, it can escape to the sides in texture voids of the surface. Otherwise, it will push its way through the whole length of the center portion of the contact area and leave the contact area at the center of the trailing edge. The fluid has then lifted the center portion of the tire contact area and reduced the net area of ground contact available for generating slip friction for braking. The same dry, powdery snow of the same moderate thickness may therefore affect the size of the net area of contact differently when present on good textured pavement versus smooth ice or a hard compacted snow base (negligible texture). Research has shown that automotive-type tires remain longest in contact with the ground through the areas along the sidewalls, as the wedge of fluid penetrates the contact area at higher velocities. Automotive-type tires are models for many friction test tires. If the different categories of friction test tires have different force systems when planing, this must be accounted for when predicting forces for an actual automotive tire configuration. Fluid Planing with Different Tire Designs A fluid planing ratio can serve as a parameter to describe the intensity of fluid planing for a given device tire configuration and surface pair. The fraction of contact patch area lifted versus the gross patch area is called a fluid planing ratio, kp. A fluid planing ratio of 0.4 means that only 0.6 of the gross contact patch remains for slip friction braking. A fluid planing ratio of zero means that the whole gross contact patch is available for slip friction braking with the ground surface. In figure E-10, a scenario with an automotive-type tire is depicted for different degrees of planing. Figure E-10. Hydro- or aqua-planing of automotive tires.

E-20 It can be argued that an automotive tire will tend to keep the line of attack closer to a vertical plane. Viscosity is a measure of the shear forces that a fluid can sustain when interspersed between opposing surfaces. Water has lower viscosity than slush or snow powder. The shear forces in the separation zone of the contact patch are therefore generally higher for snow powder than for water. The higher viscosity of snow powder also prevents it from escaping as quickly as water when there are escape voids or texture in the contact patch. Thus, the separation zone produced by the same thickness of water film and snow powder film will tend to yield a larger separation zone for snow powder than for water. In conclusion, tires are apt to lift more on snow powder than on water. The lift phenomenon is called fluid planing. Critical Planing Speed The critical planing speed is a function of fluid mass density, ρ, and contact pressure, σ, such that: ρ σ⋅= constantVC Eq. E-47 When VC is determined for a tire configuration using water, the critical planing speed for a winter contaminant, VCcontam, can be estimated using the following: γ σ ρ ρ σ ⋅=⋅= constantconstantV water ioncontaminat Ccontam Eq. E-48 where γ is the specific gravity of the contaminant. Resistive Forces from Fluid Displacement Drag on Rigid Base Surfaces The fluid displacement drag on the tire is given by the following: 221 VACF DDD ⋅⋅⋅⋅= ρ Eq. E-49 where CD is the fluid drag coefficient, ρ is the fluid mass density, AD is the tire-fluid contact area in the normal vertical plane and V is the traveling velocity. The drag force is not considered to have a significant vertical component. The fluid lift stems from the tire rolling over the fluid, which escapes under compression in texture voids or gives rise to planing. The drag coefficient is the ratio of resistance over dynamic pressure multiplied by the maximum cross-sectional area of the body, AD. There is a need to research drag coefficients for high pressure friction tester tires and high pressure heavy vehicle type tires for varying depths of contaminants.

E-21 For viscous planning of a very high pressure tire, the area AD is a constant: twAD ⋅= Eq. E-50 where w is the gross width of the tire footprint and t is the contaminant fluid layer thickness. As shown in figure E-11, the frontal area for drag can be reduced as a result of the buckling of the tire. Some contaminant may then be trapped to flow under the tire rather than be displaced to the sides. The effective area, AD, for the automotive tire type planing can be modeled as: ⎟⎠ ⎞⎜⎝ ⎛ ⋅−⋅⋅= pD ktwA 2 11 Eq. E-51 where kP is the planing factor. 0 40 80 120 160 1.0 2.0 3.0 4.0 40 80 120 160 200 Side edge (mm) Lead edge (mm) Clearance between tire and pavement (mm) Velocity = 67.7 km/h (36.5 kn) Inflation pressure = 130 kPa (18.9 psi) 0 Direction of travel a Figure E-11. Fluid film distribution in automotive tire footprint. Results of sample calculations for the drag force are shown in figure E-12. A value of CD = 0.4 has been assumed for the calculations. As can be expected, the forces are smaller for the lower density fluid and proportional to the densities. For full fluid planing, the forces are half the values of the non-planing values. This particular planing factor definition can be applied only to automotive tire types.

E-22 0 10 20 30 40 50 0 200 400 600 800 1000 Speed, V (m/s) D ra g- fo rc e, F (V ) ( N ) Water Snow k p = 0 k p = 1 kp = 0 kp = 1 Figure E-12. Sample theoretical fluid displacement drag force for the ASTM E 1551 tire on snow. SUMMARY OF MECHANICS OF TIRE–SURFACE FRICTION Given that the objective of friction measuring devices is to report braking slip friction, the dynamic influences of winter contaminants in liquid, plastic or particle form introduce errors in the reported friction values. These adverse dynamic effects contribute differently to the reported friction values for various types of devices. Generally, the adverse effects grow with increasing travel speed and the increasing deposit depth and density of the contaminate. The braking slip friction force, FB, depends on the slip speed and travel speed where adhesion and hysteresis constitute the principal mechanisms of friction. When planing occurs, the effective contact area for generating braking slip friction forces is gradually reduced with increasing travel speed. The displacement drag force, FD, depends on the squared velocity and increases rapidly with accelerating travel speed. The drag term grows and the braking slip friction term diminishes in relative and absolute terms. The rolling resistance force due to fluid lift effects, FL, will increase on surfaces with loose contaminants, as it combines with vertical components of compacting resistance forces.

E-23 The fluid forces are closely related to tire geometry, tire carcass design, inflation pressure, and weight carried by the wheel. Different tire types exhibit different behaviour with planing. The speed dependency of the error terms indicates that a measuring speed limit may exist in order to report below acceptable errors for a given surface material, contaminant type and deposit depth. In general, the error of the reported friction increases with increasing measuring speed for both a force- and torque-measuring friction device. To minimize the error, measuring should be done at low speeds if there is a possibility for significant contaminant deposit depths. When there is no significant deposit depth of a fluid or loose winter contaminant on the base surface, the difference between reported friction values from a force- and torque- measuring friction device using the same tire configuration will be the tire-rolling resistance value. MODERN TIRE–PAVEMENT FRICTION MODELS Penn State, PIARC and Rado Models The World Road Association conducted an exemplary field investigation in 1992 (Wambold et al., 1995). In a large international measurement experiment, wet pavement was studied across a wide variety of pavement materials for highways, including some runways. The objective was to harmonize friction and texture measurement devices. Several important outcomes have been reported from that experiment. One is that macro-texture is the principal reason for the speed dependency of friction. Harmonization of the friction measuring devices participating in the experiment was achieved with the support of texture information. The harmonized friction measure was therefore proposed as a two-parametric International Friction Index (IFI): a friction number associated with a reference slip-speed value and a speed number associated with the slip-speed gradient of friction. The participating friction measuring devices measured friction at different slip values. The successful harmonization resulted when the measured friction values were adjusted to a common slip-speed value of 37 mi/hr (60 km/hr). These adjustments were calculated with an exponential equation derived from what is known as the Pennsylvania State University model. It is widely used to predict friction at speeds other than the measured speed for a surface. The model has the following form: ( ) ⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ ⋅= 00 V -V eV μμ Eq. E-52 where μ0 is the zero intercept and V0 is an exponential constant.4 Both parameters are valid for a surface only and can be determined by measuring friction at several speeds. Wambold, J.C., C.E. Antle, J.J. Henry, and Z. Rado. 1995. “International PIARC Experiment to Compare and Harmonize Texture and Skid Resistance Measurements,” AIPCR-01.04.T. 4 The original Penn State Model uses so-called skid numbers for friction coefficient and the term 100/PNG instead of V0 as used here. PNG is a percent normalized gradient.

E-24 In the derived PIARC model, the zero intercept of this equation is replaced by a constant friction value at an arbitrarily chosen reference slip speed of 37 mi/hr (60 km/hr) and another exponential term. ( ) ( ) pS SeS −⋅= 6060μμ Eq. E-53 The slip speed of 37 mi/hr (60 km/hr) was chosen as a representative median value for road vehicles during emergency braking. The value of friction at that speed is μ(60).5 The slip- speed value is a parameter of the exponential term. A second parameter of the exponential term is the so-called Speed Constant, Sp, which is closely related to measurements of macro- texture for the same surface. A sample graph produced with equation E-53 is shown in figure E-13. Slip Speed, S (km/h) 0 20 40 60 80 100 Fr ic tio n V al ue , μ (S ) 0.2 0.4 0.6 0.8 1.0 PIARC Model: μ(60) = 0.6 Sp = 150 km/h Figure E-13. A sample plot of friction model for the International Friction Index (IFI). When the IFI parameters for a surface are known, the friction value can be calculated for all slip speeds for the surface. For braking slip friction, the basic friction model of Amontons can be replaced with the following friction model: ( ) ( ) NeSF PS S ⋅⋅= −60 60μ Eq. E-54 5 The original model uses the notation F for friction coefficient instead of μ as used here to differentiate between force and coefficient.

E-25 This equation is valid for wet pavement only. It has successfully captured the commonly observed influences of texture and slip speed for a device tire configuration-surface pair. For the same surface, another device would have another set of parameters, μ(60) and Sp. Inspired by PIARC’s success, we should continue our quest for more precise friction models for other tire–surface pairs. Indications are that one mathematical model may not be able to describe all the different tire–surface pairs found on roads and highways during winter. A potential problem of extending the PIARC model to another hard surface, such as rough ice, is the lack of texture measurement devices that can be used on ice. The gradient of the friction curve cannot be determined from texture devices. Additionally, the surface rather than the tire becomes the sacrificial part of the tire–surface pair. The texture effect of a sacrificial surface is usually regarded as insignificant or nil. Another outcome of the PIARC experiment seems to have the potential to rectify this problem: combining the logarithmic friction model with variable-slip measuring techniques. One of the researchers at Pennsylvania State University who analyzed the experiment results, came up with a good fit for a new friction model. The model is an implementation of a three-parameter log-normal equation, often referred to as the Rado model. This model captures the influence of the tire design and material in addition to texture, slip speed and measuring speed. The model is valid for wet pavement as it was derived from such a database. It has the following form: ( ) 2 2 ˆ ln C S S peak C eS ⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛− ⋅= μμ Eq. E-55 where μpeak is the maximum or peak friction coefficient value measured during a controlled, linearly ramped braking from free rolling to locked wheel at a constant measuring speed. Sc is the slip speed at which the maximum friction occurred and Ĉ2 is a shape factor related to texture measurements in a slightly different manner than the speed constant, SP, of the IFI. All three model parameters are determined by measurements of the ground friction measurement device using variable-slip technique. The friction value at other slip speeds can therefore also be calculated with this model. A graphical presentation of the model is shown in figure E-14. The maximum friction value is 0.75, the slip speed at which it occurred is 12 mi/hr (20 km/hr) and the shape factor is 1.05. A notable difference between the PIARC and Rado models is found at low slip speeds. Figure E-15 shows the two graphs superimposed: the Rado model is the transient phase when the brakes are first applied up to some slip, then the PIARC model follows as the speed of the vehicle slows. The PIARC model is the steady-state value of friction. In a stopping situation, the transient part happens so quickly that only the steady-state, the PIARC Model, needs to be used. However, when antilock braking systems (ABS) are used, both models must be used to evaluate stopping and stopping distance.

E-26 Slip Speed, S (km/h) 0 20 40 60 80 100 Fr ic tio n Va lu e, μ ( S ) 0.0 0.2 0.4 0.6 0.8 1.0 Figure E-14. A sample Rado model plot. Slip Speed, S (km/h) 0 20 40 60 80 100 Fr ic tio n V al ue , μ (S ) 0.0 0.2 0.4 0.6 0.8 1.0 Figure E-15. A sample comparison between PIARC and Rado friction models. Rado Model: µpeak = 0.75 Sc = 20 km/h Ĉ2 = 1.05 Rado Model: µpeak = 0.75 Sc = 20 km/h Ĉ2 = 1.05 PIARC Model: µ(60) = 0.6 Sp = 150 km/h

E-27 The PIARC model and its IFI are primarily intended for long-term monitoring of the pavement for budgeting renewal of the surface when polished or worn to unacceptable levels. The Rado model is intended for the prediction of braking performance. Automotive ABS brake systems operate on the initial rising part of the friction-slip speed curve of the Rado model. This part of the curve is often called the tire influence segment. Beyond the maximum friction value, the curve has a surface influence segment. ABS and other automatically modulated brakes are not designed to operate beyond the maximum friction point. The braking systems operate on the tire influence segment of the Rado model friction curve. Now we have a model that can predict the braking force, F, of a single wheel at a constant travel speed: ( ) WC S S peak FeSF C ⋅⋅= ⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛− 2 2 ˆ ln μ Eq. E-56 Preliminary findings of the Joint Winter Runway Friction Measurement Program suggest that the Rado model parameters—maximum friction, slip speed at maximum friction and shape factor—are unique for a tire–surface pair. Thus, surfaces may be classified with this technique. In summary, the PIARC Model is best for use with fixed-slip devices and varying measuring speeds. The Rado Model is for use with fixed measuring speed and varying slip speed. When the two models are combined, three-dimensional models are obtained as described in the following section. Three-Dimensional Modeling of Tire–Surface Friction Since travel speed and slip speed have been treated separately by different friction models as two independent variables, it would be desirable to have a combined three-dimensional friction model including travel and slip speeds as variables. Bachmann (1998) has found that repeated runs with variable-slip devices can provide data for deriving three-dimensional friction models with travel and slip speeds as variables. As can be seen from figure E-16, the series of measurements almost constitute a surface plot of friction. The curves are measured with a treaded automotive tire on dry, concrete pavement. It is more practical to use slip ratio rather than slip speed as an independent variable to view the surface plot as a full area cover. Since the upper limit slip speed at any travel speed equals the travel speed of the device, plotting with slip speed on one axis would generate a triangular shape plot projected in the speed plane. Bachman, T. 1998. “Wechselwirkungen im Prozess der Reibung zwischen Reifen und Farbahn, VDI Reihe 12 Nr. 360.

E-28 Figure E-16. Series of variable-slip measurements with an automotive tire at different measuring speeds on dry concrete pavement [6]. Based on the characteristic shapes of the curves in figure E-16, it is conceivable that three- dimensional plots of standard types of measuring tires may be produced. This can be used as documentation of the typical friction speed characteristics of a tire as an instrument sensor for different surface types and conditions. Although research strongly indicates that braking slip friction can be presented in a three- dimensional manner as shown, researched and documented universal three-dimensional mathematical models are not yet available unless the Rado and PIARC models are combined. STANDARD TIRE–SURFACE FRICTION MEASUREMENT The International Friction Index (IFI) The World Road Association conducted the International PIARC Experiment in September and October 1992. Forty-seven different measuring systems surveyed 54 sites, encompassing a wide variety of pavement types on roads and airfields in Belgium and Spain. The systems measured 67 different parameters (33 texture parameters and 34 friction parameters). The results of the experiment were presented in Montreal in 1995. The World Road Association had recognized that methods and systems used throughout the world for measuring texture and skid resistance vary significantly, causing barriers for much-needed international information exchange and comparisons. It was necessary to convert results produced by different devices to a common scale. The PIARC Technical Travel Speed, V (km/h) Slip Ratio, λ (%) Fr ic tio n V al ue , μ (V ,λ )

E-29 Committee C1 on Surface Characteristics decided to conduct an experiment to see whether harmonization could be achieved. The data collected and analyzed enabled an international scale of friction values called IFI to be defined. The IFI is now an ASTM standard (E 1960- 98) and an ISO standard (13473-1). Since wet pavement friction is speed-dependent, the PIARC model incorporates macro- texture measurements to enable the side-force, fixed-slip, and locked-wheel types of friction measurements to be related. The IFI can be calculated from the results of any friction measurement combined with a macro-texture measurement that predicts the speed gradient of the friction. The IFI consists of two parameters: F60 and Sp. F60 is the harmonized estimate of the friction at 37 mi/hr (60 km/hr) and Sp is the speed constant. Friction values can be calculated for any slip speed. The PIARC model and IFI therefore represent universal engineering tools that are valid for braked tires interacting with wet pavement types such as those encountered on highways. It was found that friction devices could be harmonized. The reference of harmonization was the average performance of all participating devices. The average performance is represented by a mathematical equation; a decaying exponential called the Golden Curve. Each device has a calibration factor to this Golden Curve at the speed of harmonization (37 mi/hr [60 km/hr] slip speed). Calibration constants were worked out for all of the participating devices and are published in the report of the experiment. The calibration constants used with the corresponding friction devices enable the Golden Curve to be recreated for surfaces, thus allowing secondary calibrations of new equipment to be performed or friction values obtained with one device to be translated to the measuring units of another calibrated device. The Harmonization Procedure The PIARC harmonization procedure is as follows. 1. The speed constant is calculated using a texture measurement of the surface. The equation used is: TxbaS p ⋅+= Eq. E-57 where Tx is a texture measurement and a and b are harmonization constants for the texture measuring device determined in the international experiment. 2. The friction measurement is adjusted to the harmonization slip speed of 37 mi/hr (60 km/hr) using the following equation: ( ) ( ) pS S devicedevice eS 60 60 − ⋅= μμ Eq. E-58

E-30 where S is the slip speed of the measurement and μ(S)device is the measured friction value by the device. For a fixed-slip friction measuring device, the slip speed is the measuring speed multiplied by the slip ratio. 3. The harmonized friction value at 60 km/h slip speed is then calculated using the equation: ( ) ( )deviceharmonized BA 6060 μμ ⋅+= Eq. E-59 when the measuring tire has a blank tread, or ( ) ( ) TxCBA deviceharmonized ⋅+⋅+= 6060 μμ Eq. E-60 when the measuring tire tread is ribbed or has a pattern. A, B and C are calibration constants for the friction device determined in the international experiment. The calibration constants are regression constants. 4. The International Friction Index is then reported as IFI60(μ(60)harmonized, Sp). The PIARC Model can also adjust the IFI to another slip reference value using the following equation: ( ) ( ) pS S harmonizedharmonized eS − ⋅= 60 60μμ Eq. E-61 where S is the slip speed for which a friction value is desired. For instance, the IFI friction value at 90 km/h, would be ( ) ( ) ( ) pp SharmonizedSharmonizedharmonized ee 309060 606090 −− ⋅=⋅= μμμ Eq. E-62 The IFI is then reported as IFI90(μ(90)harmonized, Sp). The management beauty of the IFI is that regulations can be made stipulating IFI parameters, which are universal (i.e., no tie to a particular friction device). But a friction device must have calibration constants determined, as demonstrated by figure E-17. It is natural that they initially come with the device as part of the documentation from the manufacturer, as is the common industry practice by other instrument manufacturers.

E-31 μ(60)device μ(60) Sp Sp device Golden CurveDevice Curve Harmonization Regression 60 Slip Speed, S (km/h) Fr ic tio n Va lu e, μ( S ) Figure E-17. Calibration constants for IFI are taken at a harmonizing slip speed of 37 mi/hr (60 km/hr). The reference curve is named the Golden Curve. It is an average of all participating devices in the 1992 experiment.

E-32 LIST OF SYMBOLS AND ABBREVIATIONS A Calibration constant for the International Friction Index AD Area in the vertical plane associated with contaminant deposit displacement drag AL Contact area between a tire and a fluid AR Real contact area between a tire and a surface AS Area of shearing contact between a tire and a surface B Calibration constant for the International Friction Index CD Coefficient of displacement drag CV Coefficient of variation Ĉ Shape factor in the Rado friction model (log normal) E A force, or a sum of forces, that constitute an error term in a measured braking slip force F Force FB Force due to braking slip friction FD Force resulting from positive displacement of fluid or plastic material in the frontal area of a tire FDG Reaction force in the tire–surface area due to contaminant displacement drag FE Resultant dynamic contaminant deposit force FG Reaction force from the ground FL Lift force due to dynamic fluid viscous resistance (Petroff’s equation) FLG Reaction force in tire–surface contact area due to dynamic fluid lift or compacting lift FM The horizontal force measured by a friction measuring device at the wheel axis FMD Reaction force at a wheel axis due to contaminant displacement drag FR Force due to pneumatic tire rolling resistance FS Resultant resistive reaction force in the tire–surface contact area FW Applied vertical force on a wheel axis, equal to a device mass multiplied by the gravity constant, or a controlled, vertically applied force FX Force applied at the wheel axis in direction of the x-axis (direction of travel) I Angular moment of inertia L Length M Moment MB Applied brake moment about a wheel axis MTD Mean texture depth S Slip speed SC Critical slip speed value in a Rado friction model (log normal friction model) SP Speed number of the PIARC friction model or the International Friction Index StdErr Standard Error StdDev Standard deviation

E-33 TM The moment measured by a friction measuring device about the wheel axis TX Texture measurement, generic V Travel speed VB Tangential speed of a braked wheel in the tire–surface contact area VC Critical planing speed V0 Speed constant a 1) Horizontal distance between a point of application of the vertical ground reaction force and vertical line through the wheel axis 2) Calibration constant for texture measurement with the International Friction Index 3) Zero intercept parameter for exponential friction model 4) Longitudinal acceleration b 1) Horizontal distance between vertical through the wheel axis and point of application for a dynamic lift force 2) Calibration constant for texture measurement with the International Friction Index 3) Speed parameter for an exponential friction model c Vertical distance between a tire–surface contact plane and a point of application for a resultant dynamic contaminant deposit force g Gravitational acceleration constant kL Dynamic fluid lift coefficient kP Fluid planing factor kPB Braked wheel planing ratio kPL Dynamic fluid lift coefficient including deflected tire radius and tire–surface contact area kVD Vertical displacement factor l Length of a tire–surface contact area n Number of data points or measurements, sample size r Deflected tire radius t Contaminant deposit thickness w Width of the tire–surface contact area α Angle λ Slip ratio, S/V γ Specific gravity μ 1) Friction coefficient as the ratio of a horizontal force to a vertical force in the tire–surface contact area. 2) A reported friction value. μ10, μ100 Average coefficient of friction over a 10 m or 100 m measured distance μB Braking slip friction coefficient, FB/FW μpeak A maximum or peak friction coefficient value in a variable-slip measurement

E-34 μR Tire-rolling friction coefficient, FR/FW ρ Contaminant fluid or particle mass density σ Normal stress or contact pressure σS Normal stress in the shear area of a tire–surface contact patch τult Ultimate shear stress of a surface material ν Dynamic viscosity ω Angular velocity ωB Angular velocity of a braked wheel ABS Antilock Braking System ASTM ASTM International IFI International Friction Index ISO International Organization for Standardization JBI James Brake Index JWRFMP Joint Winter Runway Friction Measurement Program NASA National Aeronautics and Space Administration PIARC Permanent International Association of Road Congresses (The organization has changed its name to World Road Association (PIARC)) SAE Society of Automotive Engineers