CHAPTER 3 BRAKING PERFORMANCE
- I ·7.-:_.. . . .
ABS test drive. (Photo courtesy ofRobert Bosch GmbH.)
BASIC EQUATIONS The general equation for braking performance may be obtained from Newton's Second Law written for the x-direction. The forces on the vehicle are generally of the type shown in Figure 1.6. Then, NSL is:
(3-1)
where:
W =Vehicle weight g =Gravitational acceleration Dx =- ax =Linear deceleration Fxf =Front axle braking force Fxr = Rear axle braking force J)1\ = A 'rodynarrli drag (.) I Jphill gradl· ( 'II!\I' I'l '.I{ \ BI{A K IN ' I'I ~ RI'O J{M/\N\I ;. H IN I ' A MI ' N'I I ,' ~ I II' VI ' I III 'I 1 I ' N!\MII ',:
TIIl',IJOIlI a,lId (l'll l hfuk illf ' 1111 ' '( 'rillS ariSl: Irllfllih 'lll('(J"l: llflh ' brak 'S where: : 1 ~ llIl~ Wllllllllhllg rt.:si~ lall ," 'rr ' 'Is, h ';trill' I'd 'lion, and uriv 'lin' drags, A x =Distance traveled during the deceleration IIlllpr,'" 'IISIV ' :I1l ;dYSIS nllh' d <..:c ' Ieralion r<": l(uir 'S detai led knowledge of all Ill!'s ' l or' 'S a 'ling un Ihe vt.: hi ck, In the case where the deceleration is a full stop, then Vf is zero, and X is the stopping distance, SO. Then:
( 'cHlshml Deceleration y2 y2 SD=_o-= - ~- (3-6) Silllpi 'and fundamental relationships can be derived for the case where it 2 Fxt 2 Dx I/'> 1l' ,I SOII ;,lbl ' to assume that the forces acting on the vehicle will be constant M 1I111111 1'. ~1 ( 1~11 a ~rake application. The simple equations that result provide an ,I PI'"' 'IOtJolllor the basic relationships that govern braking maneuvers. From and the time to stop is: 1','1 , (J- I ): Yo Yo (3-7) _ F xt _ dY t =--=- x - - -- - (3-2) s Fxt Dx M dt M w h l'lI': Thus, all other things being equal, the time to stop is proportional to the I'xt - Th 'total ofall longitudinal deceleration forces on the vehicle (+) velocity, whereas the distance is proportional to the velocity squared (i.e., V • r orward velocity doubling the velocity doubles the time to stop. but quadruples the distance required). Tid, l 'qll : lt~ ~ 11 can be ,integrated (because Fxt is constant) for a deceleration ( -,1111\1 ) 11'11\\1 1I1llial velOCity , V0' to final velocity, V( Deceleration with Wind Resistance The aerodynamic drag on a vehic1e is dependent on vehicle drag factors (3-3) and the square of the speed. To determine stopping distance in such cases, a more complicated expression is necessary but can still be integrated. To analyze this case: F (3-8) V - Yr= xtts (3-4) o M' where: IS - Till1l' for the velocity change Fb =Total brake force of front and rear wheels C = Aerodynamic drag factor .. \I ~ : :IIl S ' v 10 i(~ and distance are relaled by V =dx/dl, we can substitute Therefore: 11:1 .II II I J',q, (3-2), tnt 'grat " and obtai nih' r 'lationshi p bel ween v<..:1 (lei ty and d H, l i llll'l': S]) YdY (3-9) dx = M JO o (. -. I Yo I'IINI )AMI' N'I AI ,' (,I , VI 'III( 'I I I >YNAMJ( ',' ('IIAI ' J'b l{ J II1 This llIay h...: illt . 'r:lt .J to obtaill til' stoppin ' distallC ': The.: pararnef.l!J' "1', ," is the rolling resistance coefficient, which will be 2 discussed in the nexl chapter. Note that the total force is independent of the . M Fb+ Vo n · IO) distribution of loads on the axles (static or dynamic): Rolling resis~ance forces SI) = 2 C In I Fb J are nominally equivalent to about 0.01 g deceleratIOn (0.3 ftlsec ). Em'r'gy/Power Th ' c'lll'rgy and/or power absorbed by a brake system can be substantial Aerodynamic Drag t h l! illg a Iypical maximum-effort stop. The energy absorbed is the kinetic The drag from air resistance depends on the dynamic pressure, and is thus l I! l' II 'Y or motion for the vehicle, and is thus dependent on the mass. proportional to the square of the speed, At low speeds it is negligible. At 2 (3-11 ) normal highway speeds, it may contribute a force equivalent to about 0.03 g (1 I ':n(~rgy =~ (V0 - V f1 ftlsec2). More discussion of this topic is presented in the next chapter. Til ' power absorption will vary with the speed, being equivalent to the hl':lk ill ' force times the speed atany instant oftime. Thus, the powerdissipation is ~ n' at cs t at.lhe beginning ofthe stop when the speed is highest. Overthe entire Driveline Drag ,'.llIP. III ' average power absorption will be the energy divided by the time to '. llI)" Thlls: The engine, transmission, and final drive contribute both drag and inertia V 2 effects to the braking action. As discussed in the previous chapter on Power = M ~ (3-12) Accel erati on Performance, the inertia ofthese components adds to the effecti ve 2 ts mass of the vehicle, and warrants consideration in brake sizing on the drive wheels, The drag arises from bearing and gear friction in the transmission and Cal '1I1atioJl o rthe power is informative from the standpoint ofappreciating I hI' IlI'l J'ol'lllan<.;e required from a brake system. A 3000 Ib car in a maximum differential, and engine braking. Engine braking is equivalent to the "motor ("111111 Slt)P frolll HO mph requires absorption of nearly 650,000 ft-Ib of energy. ing" torque (observed on a dynamometer) arising from internal ~riction a~d air II st()ppl!d in 8 seconds (10 mph/sec), the average power absorption of the pumping losses. (It is worth noting that the pumping losses disappear If ~he brak 'S <.luring this interval is 145 HP. An 80,000 Ib truck stopped from 60 mph engine is driven to a speed high enough to float the valves. Thus, engme Iypi 'ally involves dissipation at an average rate of several thousands of braking disappears when an engine over-revs excessively. This can be a Iu lI'scpower! serious problem on low-speed truck engines where val ve float may occurabove 4000 rpm, and has been the cause of runaway accidents on long grades,) On a manual transmission with clutch engaged during braking, the engine braking BI{AKING FORCES is multiplied by the gear ratio selected. Torque-converter transmission.s are designed for power transfer from the engine to the driveline, but are relat~vely The forces on a vehicle producing a given braking deceleration may arise ineffective in the reverse direction; hence, engine drag does not contrIbute Ifllll1 a number of sources. Though the brakes are the primmy source, others substantially to braking on vehicles so equipped. will he discussed first. Whether or not driveline drag aids in braking depends on the rate of deceleration. If the vehicle is slowing down faster than the driveline compo I{ulling Ucsistance nents would slow down under their own friction, the drive wheel brakes must Rollin' r 'sistan " alw:IYs OPpOSl'S vdli ' 1' 1I1(ltioll ; II n ' -" it aids t.he pick up the extra load of decelerating the drivelin~ during t.he ?raking IlI:lk ' ,~ , The rollilll' Il'Si , I:II H'l' (()I ('l' ~, wi ll Iw : maneuver. On the other hand, during low-level deceleratIOns thedrIvelme drag may h' sufficient to d celcrate the rotating driveline components and contrib ( \ I ) IIll' In tIll' hr:lkil1l', '('('orl nn th 'driv' wh eoel s as well. I ~ ,I') ( ~l"Ud(' Hrak(' Factor Road 'fad' wi II l:ontri bute uin;cLly tu the braking <.:Ilort, either in a positi ve Brake factor is a mechanical advantage that can be utilized in drum brakes ~ , t.: I1 ~ · (uphill) OJ' negative (downhill), Grade is defined as the rise over the run to minimize the actuation effort required. The mechanism of a common drum (WII i 'al over horizontal distance), The additional force on the vehicle arising brake is shown in simplified form in Figure 3.2, The brake consists oftwo shoes pivoted at the bottom. The application of an actuation force, P a'. push~s the If IIII! t~r~ld " Rg, is given hy: lining against the drum generating a friction force whose magnItude IS the R, = Wsin8 (3-14) normal load times the coefficient of friction (fl) of the lining material against I'll slJIall angles typical of most grades: the drum. Taking moments about the pivot point for shoe A: (3-15) - (radians) == Grade =Rise/run L Mp =ePa + n fl NA - m N A =0 R.., =W sin e == W 8 where: e =Perpendicular distance from actuation force to pivot '1'1111,-; a grade or 4% (0,04) will be equivalent to a deceleration of ± 0.04 g N =Normal force between lining A and drum ( I ,. fI/s c2). n A =Perpendicular distance from lining friction force to pivot m =Perpendicular distance from the normal force to the pivot UI{AKES The friction force developed by each brake shoe is: I\utofilotiv brakes in common usage today are of two types-drum and and \11 I ' II. , . I as shown in Figure 3.1. Then equation (3-15) can be manipulated to obtain: F A _ ~ e and F B _ ~ e (3-16) 'tII) '- , ,,~&,,~ Pa - (m - ~ n) Pa - (m + ~ n) - ... . ~ I ,,'I I. e ~,'/: \' '/ ~' ~ .... IlN /' !1 , I, I L rum. brake and disc hrake, (Photo courtesy ofChry,\'[er Corp.) + e Ilisiori 'ally, drum hrak 's have s n common usage in the U.S . because of Ilwir high hrak 'faclor and Ih' 'asy incorporalion ofparking hrakc features. On IIII' 1Il'J',aliv 'sid', drlllll hrakes lIlay nollw as consisl 'nl inlorqu' p 'rfonnanc ,I~, disc hr:l kl's. Thl' IUWl'I' hrakl' farlllrs or di s ~ ' hrakes n''1l1ir~' hil',h 'r a 'Illalion 1'111111, :11111 d('v('lul'lIlI'lIl III' illll'/',Iall'all\illl'. hI' Ikl' i'I'II II1Il'S has hl'l'n n''1l1ifl~d Iwlllll'tli,',1 111 :11., \'1, l'llidd hi' 1I ',I'd III all wlll'(' 1 p"·,ilillll,., Fig. I. Forct's (/(,tillg (III tilt' shol's oj(/ siml)/I' dru/ll hrakl', ' ,1) ( 'IIAn!iR .;I IIRAKINli Jl E RI'URMAN ~' I ' , H INI IAMI ',NTAI ,' CII , VUIII 'I I', I I YN AMH ', ' '1'It, shu' UII Ill' ri 'hi is a "Il:adill ," sho', '1'11'1110111'111 produ "d hy til ' Drum rake Disc Brake II iL: l ion I'm' , (111 th e;: shoe acts 10 rotate il agaillsl lilt: drulll and in <': J'(.: as' th ' II i ' I i, HI for 'c developed, 'rhis "self-st:rvo" action yields a mechanical Itlvallta !, ' dtaraetcril'.ed as the "hrake factor," The brake factor is not only Q) Q) ::J pi opoll ional to 1.1 in the numerator, but is increased by its influence in the ::J ~ ~ o d"1I01lIillator. (The expressions become more complicated with lining distrib o I I- IIl r d ,IV 'J' a larger arc, but show the same effect.) Clearly, if!l gets too large, III\' 1\'1'111 "p n" may equal "rn" and the brake factor goes to infinity, in which case Velocity Temperature II Il' hl :lk ' will lock on application, Effect Fade Silo ' IJ is a trailing shoe configuration on which the friction force acts to Time Time jl'dll ',' tile application force, The brake factor is much lower, and higher Ilflpli ':It ion forces are required to achieve the desired braking torque, Fig, 3.3 Inertia dynamometer torque measurements, By using two leading shoes, two trailing shoes, or one of each, different h!'lI k ' fa 'tors can be obtained, The duo-servo brake has two leading shoes operation, The torque normally increases almost linearly with the actuation ' 1111 P k'd to r 'ther 1.0 obtain a very high brake factor. The consequences of using effort, Pa' but to levels that vary with the speed and the energy absorbed Ir i/"h hrak ' factors is sensitivity to the lining coefficient of friction, and the (through the temperatures generated), Thus: 1'() ~H i hilit y of lIlore noise or squeal. Small changes in !l due to heating, wear, (3-17) II I 1l11H' r I'll 'tors cause the brake to behave more erratically, Since disc brakes Tb = f (Pa' Velocity, Temperature) 1. Il 'k thi s st.:/J'·acluation effect they generally have better torque consistency, Efforts to model brakes by a general equation including each of ~he .111111111 1\ 1r ilt th' cost of requiring more actuation effort, independent factors and the interrelated effects results in ,a torque equatIOn '1'111': dill 'renee between the two types ofbrakes can usually be seen in their which may require up to 27 coefficients, Because the e~uat~on ~e~ends on the II "'1111' prop 1'1 iL's during a stop, Brake torque performance can be measured in brake temperature, which increases during a brake apphcatl~n, It IS necessary tlrl' Inhoratory using an inertial dynamometer, which is simply a large rotating to incorporate a thermal model of the brake in the calculatIOn process [11], III L1S S atl a 'hed to the drum with provisions to measure the torque obtained, The Experience at The University of Michigan i~ trying to model brake torque Irl'nk ' is applied with a constant actuation force to stop a rotating inertia performance in this fashion has been only partl~lIy successful. For, moderate 1I01llillaily equivalentto the mass carried atthe wheel on which it might be used, level applications, good predictions can be obtamed, H?wev~r, a high-energy Tlrt'lorqll ' measured during the stop typically looks like that shown in Figure 3,3, application (in which the temperature gets above 650 F) WIll perma~ent1y change the brake such that a new set 'Of 27 coefficients must be determmed, 011 drum brakes, the torque will often exhibit a "sag" in the intermediate 1"111 ion or the stop. It has been hypothesized that the effect is the combination The torque produced by the brake acts to generate a braking force at the III 1l'IIlp'mtme fade and velocity effects (torque increases as velocity de ground and to decelerate the wheels and driveline components, Then: I I(' as 's), Disc brakes normally show less torque variation in the course of a ',III P, Willi an excess of these variations during a hrake application, it can be (3-18) dilll 'lilt to maintain the proper balance between front and rear braking effort dill illg a maximum-effort stop, Ultimately this can show up as less consistent d"l:t.:/ 'ralion r 'rforman " in hraking maneuvers resulting in longer stopping where: d is l:III' 'S l(il, r ::: Rolling radius of the tires TIll' 111"1"" fllllllllll' hlUk l' ( ' /111 hI' IIHllklcd frolll 1111' ('lI rv('s sll l: h as sh()wn Iw = R(ltal.ional int.:rtia of wheels (and drive components) (1 R(llatillnal d" \eralinl1 or wheels III F il'llI l' \ . \. hili ,':III Ill' dill II 1111 III 1',,'dil'l .II 'I'IIIal,'1 IIV,'I' ;111 (' llIlllilillllS or w " J H INI IAMI',N'J'AI.. ' I II , VI 'IIII 'I I , IlYN A MH',' ' ~X (,' ~'pl dill ill ~ u wh'" 10 'klfp pro ' 'ss, IX is rc!al'd tll Ill, d 'C 'I 'ralion of w Both adh 'siv l.: HIIU hyst -retic friction depend on some small amount of slip 1111 ' Vdli .' 1 'tI~rou g h th '~ ' adills o/'Ih' wh 'l: 1( , Til 'r' ,arl:"IWO primary mechanisms responsible for friction coupling as Tire II ll1 slraled 10 h gure 3.4, Surface adhesion arises from the intermolecular bonds Iwi W "II th t;: rubber and the aggregate in the road surface, The adhesion ~~ l" 111I)l1>lI .' nl is the larger of the two mechanisms on dry roads, but is reduced ,1I),slullllally when the road surface is contaminated with water' hence the loss I Contact I III I ri r l ion on wet roads, ' , Len9th Tlu- hulk hysteresis mechanism represents energy loss in the rubber as it o Vertical iI/ lo," I1 S, wh '11 sli~~ng over the aggregate in the road, Bulk (or hysteretic) Load I I i\ 111111 IS nol so aff ected by water on the road surface, thus better wet traction l2i I ~ wld·v 'd wilh tires that have high-hysteresis rubber in the tread, I "" Friction ~ Force If\ Relative RUBBER ~SliP Fig 3.5 Braking deformations in the contact patch. Because of these mechanisms, the brake force and slip are coexistent. Brake force (expressed as a coefficient FxlFz) is shown as a function of slip in Figure 3.6. Slip of the tire is defined by the ratio of slip velocity in the ~ontact patch (forward velocity - tire circumferential speed) to forward velOCity: ' V - (Or SI lp= V (3-20) where: /<'11.' . I.-I AI,','I1(/IIHlI/l "I lI lt' l O rl'/ " I, (1"11 III. v =V hide forward velocity (I) 'fir' rnialicliial sp' d (radians/sec) l'IINn MI 'Nl I , I ' V I lire ( 'IIAl'llll{ J • B){AKINU PEKl"UI(Mf \ 1'1 L.t::, 0 ,11 (nllalion Pressure Q) On dry roads. peak and slide coefficients are only mildly affected by '0 0,6 !E inflation pressure. On wet surfaces, inflation pressure increases are known to Q) oo significantly improve both coefficients. 0> 0.4 - "'/- -\---+ 30 Ih C "'Ph .x. ro Vertical Load m0,2 HI---t----l-----l---I--~ Increasing vertical load is known to categorically reduce normalized traction levels (F IF ) under both wet and dry conditions. That is, as load Hysteresis x z .' 1 o increases, the peak and slide friction forces do not mcrease prop~rtlonate y. o 20 40 60 80 100 Typically, in the vicinity of a tire's rated load, both coefficients Will decrease Wheel Slip (%) on the order of 0.0 I for a 10% increase in load. Fig, J.b Braking coefficient versus slip [4]. EXAMPLE PROBLEMS The brake coefficient deriving from adhesive and hysteretic friction 1) Consider a light truck weighing 3635 lb, performing a full stop from 60 in 'J' '~I~es with slip up to about 10 to 20% in magnitude depending on IImlillOns. Under wet road conditions, the adhesive friction contribution is mph on a level surface with a brake application tha~ dev~lops a s~eady brake Jilllillishcd such that the overall coefficient is lower. The peak coefficient is force of 2000 lb. Determine the deceleration, stoppmg distance, time to stop, energy dissipated and the brake horsepower at initial application and averaged II k 'y property, usual.ly denoted by ~p ' It establishes the maximum braking IIII ' ~ • that can be obtamed from the particular tire-road friction pair. At higher over the stop. Neglect aerodynamic and rolling resistance forces. I' lip. t.lle .coefficient diminishes, reaching its lowest value at 100% slip, Solution: I 'Pl' 'sentmg the fullloc~ condition and denoted by ~s' In a braking situation, lip 'orresponds to the highest brake force that can be generated, and is only The deceleration may be calculated from NSL: 11,1 'nret,ically possible to achieve because the system is unstable at this point. 2 I 'ur ,I given brake torque output level, once the wheel is decelerated to achieve _ F x _ Fb _ (2000 Ib) 32.2 ftlsec =17 72 ~ Dx - M - M - 3635 Ib . sec 2 pp" any disturbance about this condition results in an excess of brake torque wlllch causes further deceleration ofthe wheel. The increased slip reduces the The deceleration can be computed directly in terms of g's by using the hmk . force such that the wheel deceleration continues and the wheel goes to III 'k, nly a brake release (as in an anti-lock control) can return the wheel to equational form: mph fil' 'J'ilti on at ~p' Fx Fb 20001b 0.55 g 12.08 sec Dx (g) = w = W = 3635 Ib = = In addition to the tire and the road as key elements in determining the II i 'lion coupling available, other variables are important, as follows. Now that the deceleration is known, the stopping distance may be com puted (Eq. (3-6)): VelocHy V 2 V 2 (3-6) On dry roads, both peak and slide friction decrease with velocity. Under SD= _ o- =_o wei 'ol1d il ions, even greater speed sensi ti vity prevai Is because of the difficul ty 2 Fxt 2 Dx M III displa in ~ waler in the contact patch at hi gh speeds. When the speed and 2 w IIl'rI~ i l.n11 hi 'k n 'ss ar' sllfl'i 'i 'nt. the lir 'Ir 'ad will lift from the road creating = (88 ft/sec) = 21R.Sl ft I t'lImllllllll kIlIlWIl as hydroplanillg, ") (17.72 fIls' 2) •• " ,, '(\lVII ' I,/ 1 1\1 .... III ' 'IVNI\MII ',': V""" ', " r 1'I IAl n;){ \ . UKAKIN I rERFORMJ\N l.: '('It, (illl!' 1(1 Sl(,p '( Hlll' S "(1111 I ':q, ( ~\ 7): Thus, roughly 4 feCI wi II be cut from the stopping distance when aerody _ Yo _ 8~ fUs ee namic drag is included in the calculation. The drag itself is only 74.4lb at the (s - F IM - = 4.966 sec beginning of the stop and decreases with the square of the velocity, so its xl 17.72 flfscc 2 contribution becomes much less during the course of the stop. 'I'll, energy dissipated comes from Eq. 0-1 J): 2 (Yo - Y 3635lb (88 ftfsec) 2 I ' ~ncrgy =~ f2y = FEDERAL REQUIREMENTS FOR BRAKING 2 (32.2 ftfsec 2) PERFORMANCE =437,103 ft-Ib Out of the public concern for automotive safety in the 1960s, the Highway '1'11.(' power dissipation at the point of brake application is simply the brake Safety Act of 1965 was passed establishing the National Highway Traffic l or ' . lImcs the forward velocity, which is: Safety Administration charged with promulgating performance standards for Powcr (initial) =(2000 Ib) 88 ftfsec = J76,000 ft-Ib/sec new vehicles which would increase safety on the highways. Among the many standards that have been imposed are Federal Motor Vehicle Safety Standard IJP (initial) = (176,000ft-Ib) 1 hp -320h (FMVSS) 105 [5], establishing braking performance requirements for vehicles sec 550 ft-Ib/see - P with hydraulic brake systems, and FMVSS 121 [6], establishing braking performance requirements for vehicles with air brake systems. ( III average over the stop, the power (from Eq. (3- I2» is: FMVSS 105 defines service brake and parking brake performance require Y 2 2 "ower =M _ 0_ = 3635lb (88 ftfsec) ments over a broad range of conditions, such as: 2 ts 2 (32.2 ftfsee 2) 4.966 see • Lightly loaded to fu1Jy loaded at gross vehicle weight rating (GVWR) _ 437,103 ft-Ib ft-lb • Prebumish to full burnish conditions - 4.966 see =88,019 see = 160 hp • Speeds from 30 to 100 mph . 2) For t/~e vehicle described in the previous problem, calculate the sto in • Partially failed systems tests d,:"tan <.;e ~akmg aerodynamic drag into account. The aerodynamic drag~~rc; wJlI h given by: • Failure indicator systems 2 Fa = C y2 = 0.00935 (lb-see ) y2 ( ft2 ) • Water recovery 2 2 ft sec • Fade and recovery Tht.: stopping distance may be computed from Eq. (3- J0): • Brake control force limits S - MIl(F b + C Yo 2y (3-10) - 2 e n] Fb 2000 Ih + n.00935 Ib-sec 2 (88 ft ) 2 Although the standard is quite detailed and complex, the requirements for ::163 5 Ib In ft 2 sec stopping distance performance can be summarized into five tests: (Cl .OOI) . r Ih-st' 2 ) ( 2.2 fr 2000lh ,., - s .. J ) Pi rst effecti veness-A fu II y loaded passenger car wi th new, unburnished brakes must be able to stop from speeds of 30 and 60 mph in distances that ' 01'1' 'sr ond to average decelerations of 17 and 18 ftfsec2, respectively. 59 CHAfYrER 3 - BRAKING PERFORMANCE FUNDAMENTALS OF VEHICLE DYNAMICS 2) Second effectiveness-A fully loaded passenger car with burnished 1 appropriately for the foundation brakes installed on the vehi~le. Proportioni~g Ilrakcs must be able to stop from 30, 60 and 80 mph in distances that correspond then adjusts the brake torque output at front and rear wheels m accordance With til average decelerations of 17, 19 and 18 ftJsec2, respectively. the peak traction forces possible. 3) Third effectiveness-A lightly loaded passenger car with burnished The first-order determinants of peak traction force on a? axle ~e the hrakt.:s must be able to stop from 60 mph in a distance that corresponds to an instantaneous load and the peak coefficient of friction. Durmg brakmg, a 2 dynamic load transfer from the rear to the front axle occurs. suc.h that the load :IV 'rage deceleration of 20 ftJsec . on an axle is the static plus the dynamic load transfer contnbutlOns. Thus for 4) Fourth effectiveness-A fully loaded passenger car with burnished a deceleration, Dx: hrakc.~ s must be able to stop from 30, 60, 80 and 100 mph in distances that 2 1.' 01'1' 'spond to average decelerations of 17, 18, 17 and 16 ftJsec , respectively. (3-21 ) 5) Paltial failure-A lightly loaded and fully loaded passenger car with a failure in the brake system must be able to stop from 60 mph in a distance that and 2 'orr 'sponds to an average deceleration of 8.5 ftJsec . (3-22) W r =~ W - ~ ~ Dx =W rs - W d " is notable that the hydraulic brake standard (FMVSS 105) has stopping di!'tall 'c requirements only for dry surfaces of an 81 Skid Number. (Skid NlIlIlh 'r is the tire-road friction coefficient measured by American Society for where: 'Il'still /.! and Materials Method E-274-85 [8]. Although the Skid Number is Wfs =Front axle static load IJII ' ilslln~J with a special, standard tire, the Skid Number and coefficient of W Rear axle static load II il,tillil ar gem.:rally assumed to be equivalent.) The air brake vehicle standard rs = (I ,MY, ~ J 2 1) has stopping distance performance requirements on both wet W d =(h/L) (W/g) Dx =Dynamic load transfer :,111 I'Ll ' 'S (30 Skid Number) and dry surfaces (81 Skid Number). Obviously, the llllldl;nt hrakc system designerconsiders a range ofsurface friction conditions, Then, on each axle the maximum brake force is given by: 1'111111 at least 30 to 81 SN, despite any gaps in the Federal performance standards. hW (3-23) Fxmf= ~p Wf= ~p (Wfs + [g- Dx) HI{AKE PROPORTIONING and hW (3-24) Till' hraking decelerations achievable on a vehicle are simply the product F xmr= ~p W r= ~p (W rs - L g Dx) III appl ication level and the brake gains (torque/pressure) up to the point where III will occur on one of the axles. Lockup reduces the brake force on an 'kllp where: lIxl .. and results in some loss of ability to control the vehicle. It is well !l = Peak coefficient of friction I I: '0 rnizcd that the preferred design is to bring both axles up to the lockup point p . ~ ill\ultaJl 'ollsly. Yet. this is nol possible over the complete range of opC'rating The maximum brake force is dependent on the decelerati.on, varymg n llllliliom; 10 which" v 'hi 'I 'will h' ·xposed. Balancin r the brake outputs on differently at each axle. Figure 3.7 shows grap~ical1y the maximum brake bllih Ihl' froIII alld r('ar axl 's is a 'hi 'vl'll hy "prLlporlioning" Ih pr>ssllrc fore s according to the above equations for a tYP.lcaJ passenge~ car ~n both,a hi gh and low '0 ffi icnl surface. T~e Jecderatlo~ IS shown m Units of g s kqllivalr 1)/1'')' I\tI('JIIpls al hraklllj!. on an axle tlhove the boundary value I 11111111'111 I . ' iel', III I ' I"" " '~~ III 11" 11 I H'W 1'1 ,11., ,,.1111 ' " W fllllllI" by 11'(11'1111 '" ",.I ~l' 1I l'l'llI'lIl iolll, lllill ", 111.11111 ' I" I' I" '" I dillo d, 1111' dill 1111 ,1111,,111" I(' slIlt :. i ll "!l'I-IIP 1111 IIII' I k . II! I'UNDAMENTALS OF VEHICLE DYNAMICS CHAPTER 3 - BRAKlNG PERFORMANCE Thus the maximum braking force on the front axle is dependent on that ____ 2000~ ----~Fr~o~nt~------present on the rear axle through the deceleration and associated forward lo~d :0 transfer resulting from the rear brake action. Conversely, the same effect IS -Q) 1500 evident on the rear axle. These relationships can best be visualized by plotting ~ - /lp=0.81 o Rear the rear versus front brake forces as shown in Figure 3.8. lJ.. ~ 1000 ro~ Front 500 - - - /lp= 0.30 - - -Rea7 - - - - _ Ilf/L Slope 1 Ilph/L Fig. 3.7 Maximum brake forces as a function ofdeceleration. - Ilph/L ;Q 1500 .---- Slope =-- 1 + IlphiL ~ ~ Inasmuch as the equations above contain the deceleration as a variable, Q) I h 'y do not provide an explicit solution for the maximum braking forces on an ~ cO IIxl . These can be obtained by recognizing that the deceleration is a function 'E 1000 ofth . t.otal braking force imposed on the vehicle (neglecting for simplicity the u..e (lllt'r forces that may be present). To solve for Fxmf' we can use the f 'lationship: (3-25) Rear Brake Force (Ib) Dx = (F xmr + F xf) M (3-26) Fig. 3.8 Maximum braking forces on the front and rear axles. Substituting into Eqs. (3-23) and (3-24) yields the following equations for III· maximum braking force on each axle: The horizontal axis represents rear brake force, which is generally propor tional to the rear brake pressure (related by the torque-to-pressure relationsh~p IIp(Wfs+ h Fxr) for that foundation brake). The vertical axis is front brake force, agam L proportional to front brake pressure in accordance with the br~e gain. The Fxmf= ----=h- 0-27) I - IIp L origin of each line is obtained from Eqs. (3-27) and (3-28) by settmg the brake force of the opposite brake to zero. P p ( W J's :~ ,; X r) Lines for the maximum front axle brake force slope upward and to the right ,; 1111 (positiv ') at the slope ofJ.tp h/U( I - ~ h/L). L.ines f~r the rear axle ~aximum J I ,I II hrakl' fOfCl' slope downward III th ng~1 (n gallvc) With a slope that IS equ~1 to PI "" 11/1 . I( II lip /)/1 .), 111l ' II ':Js iIW IIII' sIIIfHCl' CIl·f{j 'i 'nt or the CG h 'Ight I·I INI)/\MI 'NTI\I .S n l ' VI'III 'I I ' D YNI\MH 'S perfo rlllance objcetives for low coefficient surfaces should be included by the ill '1' 'a ~ 'S tit' slopes o f th' maximulIl brake force lill cs on the graph. Varyill r thl' load 'orHJition on thc vehicle translates the origin ofeach of the lines on thc brake designer as well. To date, low coefficient performance has only been l' ltl ph . The intersecti on point for the front and rear brake boundaries can be specified in FMVSS 121, which defines braking performance requirements for d ,t 'I'luincd by manipulating equations (3-27) and (3-28). Designating the air-braked trucks. poillt s as Fxl'i and l\ri, it can be shown that these coordinates are: The primary factor determining brake proportioning is the gain of the brakes used on the front and rear wheels. The brake force on individual wheels (3-29) can be described by the equation: Pa Fb= Tb=G (3-31) (3-30) r r ;\n attempt to brake the vehicle to a level that goes above the front brake where: lor ' . boundary will cause front wheellockupto occur, and steering control will Fb = Brake force h ' I( 1St. Likewise, braking effort that falls to the right ofthe rear brake boundary ' H II S ~ S rear wheel lockup, which places the vehicle in an unstable condition. Tb = Brake torque Th' ins t. ability has safety implications and therefore warrants careful consid r =Tire rolling radius ('r:lt inn in the design of the brake system. This issue is discussed in more detail II iI lat 'I' section. G = Brake gain (in-Ib/psi) I n a graph of the form of Figure 3.8, the deceleration is proportional to the Pa = Application pressure 1' 11111 () I' the front and rear brake forces. Thus 2000 lb offront brake force with I' l' l fJ r 'IIJ' force, 1000 Ib front with 1000 Ib rear, and zero front with 2000 Ib rear hI nk ' fo rce, all correspond to the same deceleration level, and a line ofconstant Achieving good performance overthe full range ofconditions under which dl' ' 'Ieration can be plotted by connecting these points. Ifthe same scale is used a vehicle operates can be difficult. As an example of the complexity that can fO l' the front and rear brake forces, lines of constant deceleration plot as 45 be experienced in trying to identify appropriate brake proportioning, consider d 'gr 'c diagonals on the graph. the case shown in Figure 3.9. The figure illustrates the range of variations that 2 arise from vehicle loading (lightly loaded and GVWR) and surface friction (30 If a deceleration capability of 20 ftlsec is required on the 0.81 ~ surface, and 81 SN). On the graph the brake force boundaries and deceleration Ill y 'o mbination offront to rear brake force would satisfy that requirement so requirements have been plotted for the FMVSS 105 dry surface tests c??di It UI 'as it falls in the triangle bounded by the deceleration line and the maximum tions, In addition, similar boundaries have been plotted for wet road condItIons Iu ake force lines for the 0.81 ~ surface. assuming a friction coefficient of 30 SN. Under the wet conditions, a 2 "Brake proportioning" describes the relationship between front and rear deceleration performance goal of 8 ft.lsec (0.25 g) has been assumed. III ak" forces determined by the pressure applied to each brake and the gain of To achieve all performance goals, a proportioning design must be selected ('n 'h. it is represented by a line on the graph starting at the origin and extending that passes through all of the triangles shown. This cannot be achieved with a Il pward and to the ri ght. ;\ fixed, or constant, proportioning is a straig ht line. straight line providing a constant relationship between front and rear brake Th pri mary chall cng , in hrak syst 'm d 'si n is th ' task of s 'I 'ct.ing a force. A so lution to this problem is to incorporate a valve in the hydraulic pfo('lortionin", ralio (till' slop ' ora lin ' 0 11 th 'l',r'lph) th at will satisfy a ll d si'l1 syst ~ m that hall g 's th ' pressure going to the rear brakes over some po~io~ of l'o:ds (It- spilt· tl\(' val ia hilitil' s ill Sill f: Il'l' 1'1 it: tioll, frollt/r ·ar w·i ',ht distri butioll, th '(IP 'r:lt ill l'- pi r SS Il I(' r:r ng '. Such a valve is known as a pressure proportJOnmg ( '(; II l·i/',ht , :1 1111111 :1(.. (' ( 1IIIlIII IIIII , 1\ 1111111(,(' 1 ullhl' ~,a ' 'lhjl'vtivl'S all' dl'lllll'li h v;t1Vl' , M, lsi p ll '~ , ~ ,II\( ' llI (l IH1I1 iOllin g va lves in common use today provide equal ti ll' I.' MY, 'S (( 1'1 111.1 ~ 1I1 / · :,I ,III1I ,lId 11111 11' WIIlI I', d l! 'l tiVt' llI' ss tl· st:. . :J1lhlllll',h I' " '~, ~ ,III (, til bllih I" lllt :t lld 1( ' : 11 hl:J(.. l· S lip to a . rtain pressure level, and then /,1 (," H INI IAMI 'NTAI ,S ( JI ' VJ.'llIc 'I 1' 1 IYNAMIC ', ' mance trian YI ·s <.Ill 1101 overlap in those cases, so no choice of proportioning [' ] 1S I Eflecliveness will satisfy all goal s. Several solutions are available. In Europe, load-sensing proportioning valves have been used on trucks for some years. These valves, 2nd Eflecliveness o installed on the axle(s), sense the load condition and adjust the brake propor 3rd Eflectiveness tioning appropriately. Less commonly used is the inertia-proportioning valve which senses the deceleration rate and can adjust proportioning in accordance l i'1~ 1 ~ = 0.3, lightly loaded with the deceleration level. Finally, anti-lock brake systems offer a versatile method of automatically proportioning brakes that is becoming well accepted • ~ = 0.3, GVWR in the automotive industry. 1000 ANTI-LOCK BRAKE SYSTEMS Rather than attempt to adjust the proportioning directly, anti-lock systems (ABS) sense when wheel lockup occurs, release the brakes momentarily on locked wheels, and reapply them when the wheel spins up again. Modem anti lock brake systems are capable of releasing the brakes before the wheel goes to lockup, and modulating the level ofpressure on reapplication to just hold the wheel near peak slip conditions. 1000 1500 Rear Brake Force (Ib) The concept of ABS dates back to the 1930s, but has only become truly " g, I. 'J Fmnl/rear brake force graph for multiple braking conditions. practical with electronics available on modem vehicles, An ABS consists of an electronic control unit (ECU), a solenoid for releasing and reapplying 1("(111 ' thr rate of pressure increase to one of the brakes thereafter. A pressure to a brake, and a wheel speed sensor. The ECU normally monitors Jlloportiolling valve identified as a "500/0.3" means that the pressure to the vehicle speed through the wheel speed sensors, and upon brake application IlolIl aJ~d rcar brakes is equal up to 500 psi. Above this level the pressure begins to compute an estimate of the diminishing speed ofthe vehicle, Actual Ploportloned to the rear brakes increases at only 30% of the rate going to the wheel speeds can be compared against the computed speed to determine tI (lilt hrakes. That is: whether a wheel is slipping excessively, or the deceleration rate ofa wheel can be monitored to determine when the wheel is advancing toward lockup. Pf = Pr =Pa =Application pressure for Pa < 500 psi (3-32a) Different ABS designs use different combinations of these variables to determine when lockup is imminent and brake release is warranted. At that 1'1' = Pa and Pr =500 + 0.3 (Pa - 500) for P > 500 psi (3-32b) a point a command signal is sent to the solenoid to release the brake pressure, With this proportioning it is seen that it is possible to achieve a front/rear allowing the wheel to spin back up. Once the wheel regains speed, the pressure (,1 :Ik ' balance satisfying all dry surface conditions as evidenced by the fact that is increased again. Depending on the refinement ofthe control algorithms, the th ~' prl),p0l1.ioning line passes through all ofthe performance triangles. Theonly pressure rise rate and the final pressure may be controlled to minimize cycling tllHI ( 'X 'l'r IS the fully loaded vehicle (OYWR) on the low coefficient surface, of the brakes. Wlll.-'I.e 1I.ll' brak . pr porti onill will not quite achi eve 0.25 g. Tn every 'lise, the 1'1111 III(h eal 'S Ihat frollt II) 'k ur will oc 'ur fi rs t. Figure 3.10 shows a typical plot of wheel speed cycling during the stop of a vehi cle with ABS. When the brakes are first applied, wheel speeds diminish twhirvillr }'.()(HIIIIII(\lIll illllill}' is ,'slwl'iall dil'flvllit Oil t('\lrks hl'('allsr Ill' mM ' or I 'ss ill a ' ''ordal'lt: wilh the vehicle speed in region I in the plot. Ifthe tilt' di',p:r IIl Y Ih 'lw( 'I' 1I 1II,Idl'd ,lI ld ('lIlpl y ('l1ll1ln ioll:" T picall , IIII' 111'1/111 IWa h 's :rr~' applil'd In :r hi}"h kvcl, or th' road is slippery, the speed of one or ( t/ . 1·1 IN I, MI ·N I I ',III , 1' 1111 '1 I Il'l'NA M If','\ ( " 1/\ 1',(,1,.1< .1 111< /\ J( INt; PERI ',()I 11101(' wlll"l 'l:-, ht l '.i Jl ~ , In IIl up I 1,,1111 (poill! . ). illll it'llt ill l' t"at I'" til" has '011' URAKIN(; EFFI 'lEN -'V till olln ll ~I~I ' pl:uk ortlll' p-slir 'III Vl' alld i. ., IH;;ldiJl /.'. toward III 'ku p. At this poillt I"t: AilS 1~11 l: "V 'II'S alld J' 'I 'as 's th ' i>rak ' S 011 Ihos ' wheels bcftH" lockup Rcwgnil'.ing that braking performance of any vehicle will vary according ~ '~' C II J'~ ~ pOlnl . ). 0 11 "th 'whe 'I sp ~ 'u pi cks up a ~ ain the brakes arc reapplied. to the fri ction of the road surface on which it is attempted, the concept of I" ' ~ 'h, ~ , 'IIV ' o~ th ' A I3 S, is to kecp each tire on the vehicle operating ncar the braking efficiency has been developed as a measure of performance, Braking p ' 11" nf till' p slip clirve lor that tire. This is illustrated in Figure 3.1 J. efficiency, llb' may be defined as the ratio of actual deceleration achieved to the "best" performance possible on the given road surface, ltcan be shown with the use of the equations presented earlier that the best performance any vehicle can achieve is a braking deceleration (in g's) equivalent to coefficient of friction between the tires and the road surface. That is: ""0 ,U REAR WH EEL LOCKUP Efficiency .13 In the discussion so far, wheel lockup has been considered only as a .7 boundary on braking performance. However, it has great impact on the .6 handling behavior of the vehicle as well, and must be considered by the brake designer. Once a wheel locks up it loses its ability to generate the cornering .4 forces needed to keep the vehicle oriented on the road . .3 Lockup of front wheels causes loss of the ability to steer the vehicle, and .2 it will generally continue straight ahead despite any steering inputs, drifting to . 1 the side only in response to cross-slope or side winds . O~~~--~--~~--~__~__~~__~__-L__~L-~ o 30 40 50 60 70 80 90 100 It is well recognized that rear wheel lockup places a motor vehicle in an Application Pressure (psi) unstable condition. Once the wheels lock up, any yaw disturbances (which are always present) will initiate a rotation of the vehicle, The front wheels, which FiJ.: 3. J2 Efficiency plot for a tractor-semitrailer. yaw with the vehicle, develop a cornering force favoring the rotation, and the yaw angle continues to grow. Only when the vehicle has completely "switched /lxl ' .,> would have the same braking coefficient at a given application pressure, ends" is it again stable. On long vehicles (some trucks and buses) the rotational ilHlicalin that they all brake in proportion to their load. However, the diverse accelerations are usually slow enough that the driver can apply corrective steer IUlld l'cHl(litions, longitudinal load transfer during braking, and shift of load and prevent the full rotation. However, on smaller passenger cars, it is hl'l WC'C'II tandem ax les due to brake reactions (inter-axle load transfer) preclude generally accepted that the average driver cannot readily control the vehicle in 1I('IIc' 'I harmony of the system. This is the reason the braking efficiency falls such a driving situation. Thus there is a philosophy among automotive Iwlow til· maximum theoretical value of I. In the case ofthe tractor-semitrailer designers that a front brake bias constitutes the preferred design. howl1 II 're, the braking efficiency rises quickly to a value of 0.9 at low brake IIppli 'ution pressures, but drops off again at higher pressure due to the spread The preference for a front brake lockup first cannot easily be achieved in ill hruking coefficient among the axles at high deceleration conditions. a brake system design under all circumstances because of in-use variations in brake gain, CG height (particularly on light trucks), pavement friction, and parking brake requirements. The potential consequences in the hands of the .9 motorist have been estimated using the braking efficiency as the measure of .8 Axle performance [9]. The basis for it arises from studies of driver behavior that .7 5 show that brake applications occur on the average about 1.5 times per mile . .6 4 Though most of the brake applications are executed at a moderate level, high 3 decelerations are required in a certain percentage of the brake applications. .5 2 Braking level demands of motorists are shown in Figure 3.14, which plots the 8 .4 1 percent ofdecelerations exceeding given deceleration levels. Twenty percent .3 of all brake applications exceed 0.2 g, only I % exceed 0.35 g, and less than 0.1 % go up to 0.5 g. ,1 n The comparison ofdeceleration demands in normal driving to the available 0 10 friction level ofroads is shown in Figure 3.15. The distribution of road friction 70 80 o 100 'icnts is 'stirn ..1 'd from lIumerous surveys of"skid resistance" routinely I\ppllc Ili(1/1 Pr c ..IItO (p•.I ) co"rn Illadc' hy 111;111 hil'.hw;1 (h'partrnenls. By and lar . most roads have friction ""g ( I ( /l1(/~lIIg, ",'111, 1"/1',,/1 /11 '" (11/, '\ (1/ (/ I/(il t'" ,1I 'IIII II.tI'''' , le V( "', ~. l1fjll i('lIt tll :1I I (11l11l""l al " I Ill' dC '('('kl atillll dVrllands Ilfthl' motorists if /11 It ('IIAlrI'U< , • URAKINli I'IlRI''ORMAN ' E l'I/NI )AMI ~ NTAI . S 01 : VPIII('1 I; IJYNI\MI!' ~ 100 ~--~------~ 5 Deceleration Friction "Demand" ~ ...-. 'iii 4 o o c ...... 10 Q) (/) 0 (l) 3 u £ c :0 CO (1j "0 .0 (l) 2 (l) e u a... til c .9 - Mortimer (1970). ~ Power brakes (l) (l) --Mortimer (1970). u Coefficient Manual brakes + o(l) 0.1 o Carpenter (1956) Fig. 3. J5 Comparison offriction demand and availability. + Giles (1956) x Kummer & Meyer (1968) 0.001 15 Lockup Frequency 0 .1 .2 .3 .4 .5 .6 .7 Q) u 14 1 in 3.6 weeks (1j Deceleration (g) 1:: :J 13 Fig. 3.14 Dislribulion ofbraking decelerations wilh passenger cars. Cf)..... Q) 12 ~ 11 ..: Ih' fri ction is efficiently utilized. That is, if the brake systems on all vehicles (1j Q) 10 w 'r ' 100% efficient under all conditions, little overlap would occur in the >.... 9 Q) braking "demand" and friction "available" curves, and there would be few a... 8 1.4 months CI) hrnking instances in which wheel lockup occurs. c.. :J 7 ~u Ilowever, when braking efficiency is less than 100%, higher friction is 0 6 ....J I 'qllircd to achieve a desired deceleration level. With lower efficiency the 5 '0.... "rri 'linn demand" curvE' shifts to the right. Thus the overlap and frequency at Q) 4 .0 wlti h braking demand will exceed the friction available will increase. Using E 3 lit' av 'rag ' figure of 1,5 hrake applications per mile and I o,noo miles per year :J Z 2 1.4 years fo r a Iypi 'al pass ' 11 Ter car, the frequency of wheel lockup in braking can be 3.6 years 1 n ;lilllal'd for cliff'r'lll hraking syst m -f'fi ci'll i 'S as shown in Fi 'life 3,1 6, 8.9 years ('karly, il illllsirait's lit' aCllt' St.: l1silivil y (If 10 'kllp frequency to hrakin y l' t t Ie h·IIl·Y. I t I ht' ilid Itril'Il('y is dill' It I II 11 ' 111 hi ll Sin thl' hrake for 'disl..ihlll ion, Braking Efficiency (%) IIII' l(ld\ lIp ~, will tI\'(' 1I1 1111 1111' 1(' 1\1 II It " 111 111 d ill'l'lillllal ill slahilil willll'slIII. FIg, 3,1() l'fj'ti;('I/'{I/,,'I/w'I/('\10/'1 1".1"'/' "1','//1.1', MO',1 ill ', 1111(' 111'1 " , will III' 1111 111,111" Willi IIIW I'I lI i,'lillll kvl'ls. wh it'll !III' /1 / ' 1101111: 111 wl,'l lund l'Plldilillllc" Si ll l,'c IIll" lIIaj ol il o i" llwsl" illSI;III" s will () ' 'ur EXAMI'I,E PI{()HLEM ull I oarl s wil II rricl ion ' I wl'l i ' je nl s ill 111l' r:m/', . () f O,4 1() () f), 1);111 ielll ar '1l1pll:ISi s 'akulal' Ih -, hraking coefficients and braking efficiency for a passenger :-. lIo(lIJ h, pia ' 'd 0 11 ohLlinillg ' O(Hj hl PEUAL FORCE GAIN Weights: Wf= 2210 Ib Wr = 1864lb Total =40741b 1':1' 'oJlomics in the design of a brake system can play an important role in Front brake gain = 20 in-lb/psi Rear brake gain = 14 in-lb/psi Ihe cas' wilh which the driving public can optimally use the braking capabili Proportioning valve design = 290/0.3 Ii', hllilt illio a vehicle. Aside from positioning of the brake pedal, the effort :lIld displacement properties of the pedal during braking are recognized as Solution: Inllll'ni ial design variables. In the I 950s when power brake systems first came The easiest way to visualize the answer is to tabulate data in columns as ililo /1. 'llcraluse, there was little uniformity among manufacturers in the level shown below. The calculation steps are the following: !lld'fml and pedal displacementproperties ofthe systems. In 1970 the National Il i/' I!way Traffic Safety Administration sponsored research to determine 1) The front application pressure is the reference, so we list values from (''1 ' \ 1111 Iflli . properties for the brake pedal that would gi ve drivers the most 100 and up. (' ll n ' llv ' '()I1lrol 1101. The research identified an optimum range for pedal 2) The rear application pressure is calculated from the front using the 1III\'\'I',lIi ll I he n~lationship between pedal force and deceleration. Figure 3.17 relationship similar to that given in Eq. (3-32). Namely, howl-, Iii ' I'l'S ult,S from the NHTSA study indicating the optimal gain values by for Pa < 290 psi (3-32a) Ihl' ', IUllkd :If' a. Pr =Pa for Pa > 290 psi (3-32b) Pr =290 + 0.3 (Pa - 290) 3) The front and rear brake forces are the product of the application pressure on that brake times the torque gain times two brakes per axle divided ,065 .007 by tire radius. Pf and - -0> Fxf= 20fT 4) The deceleration is the sum of the brake forces divided by total vehicle weight (this results in deceleration in units of g). - Fxf+ Fxr Dx- W .2 5) The front and rear axle loads are calculated from Eqs. (3-21) and (3-22). Wf= Wfs + (hlL) (WIg) Ox (3-21) 60 80 100 120 140 and (3-22) P del Fo re (Ib) W r = Wrs - (h/L) (WIg) Ox wI! ' r ' "I x" is in unil s or ft/scc2 . I I I ' 1I A I ' l'I ~ I' \ I TV\ K IN I 1'1 ', HI {ll~MANI ~ H INIIAMI ~ NTAI ,S 1)1 ' VI'IIII 'I I, IIYNAMII ',' III 'Ill alld )( 'adin ss O llllll;tud, Alexandria, V A, December 1976, (,) 'I'll' hrllkill' '0 'l'Ii ~ i '1I1s (fll alld PI') aI" III' ralio lIfaxlc hrak' force HI ~I XI load, 252 p. .. r ~I _ x and 4. Meyer, W.E., and Kummer, H.W., "Mechanism of Force Transmis r- Wf sion between Tire and Road," Society of Automotive Engineers, Paper No. 620407 (490A), 1962, 18 p. 7) The braking efficiency, llb, is the deceleration divided by the highest 5. "Standard No. 105; Hydraulic Brake Systems," Code of Federal Ill' Ih ' Iwo braking coefficients from the axles, Regulations, Title 49, Part 571 .105, October 1, 1990, pp. 199-215. P1' P Ff Fr Dx Wf Wr r ~ fir Tlb 6. "Standard No. 121; Air Brake Systems," Code of Federal Regula I()() psi 100 psi 3301b 2311b .138 g 23161b 17581b .142 .131 97% tions, Title 49, Part 571.121, October I, 1990, pp. 366-382. 200 200 661 462 .276 2422 1652 .273 .280 99 7. Gillespie, T.D., and Balderas, L., "An Analytical Comparison of a European Heavy Vehicle and a Generic U.S. Heavy Vehicle," The ~ 00 293 991 677 .409 2525 1549 .393 .437 94 University of Michigan Transportation Research Institute, Report 400 323 1321 747 .508 2601 1473 .508 .507 100 No. UMTRI-87-17, August 1987,374 p. . 00 353 1651 816 .606 2676 1398 .617 .583 98 8. "Test Method for Skid Resistance of Paved Surfaces Using a Full Scale Tire," Method E274-85, 1986 Annual Book of ASTM Stan (,()() 383 1982 886 .704 2752 1322 .720 .670 98 dards, American Society for Testing and Materials, Philadelphia, 700 413 2312 955 .802 2827 1247 .818 .766 98 PA. 9. Ervin, R.D., and Winkler, C.B., "Estimation of the Probability of NIlI's: Wheel Lockup," IAVD Congress on Vehicle Design and Compo a) The braking efficiency starts high (97 - 99%) by the match ofthe brake nents, Geneva, March 3-5,1986, pp. DI45-DI65. gains and axle loads, but begins to diminish with deceleration because of the 10. Mortimer, R.E., Segel, L., Dugoff, H., Campbell, J.O., Jorgeson, d ~ 'r '<'sing load on the rear axle. C.M., and Murphy, R.W., "Brake Force Requirement Study: Driver h) When the application pressure reaches 290 psi, the proportioning valve Vehicle Braking Performance as a Function ofBrake System Design "ki 'ks in" reducing the pressure rise rate on the rear axle. This brings things Variables," The University of Michigan Highway Safety Research h:lck into balance providing 100% efficiency at 400 psi. Institute, Report No. HuF-6, April 1979,22 p. 11. Johnson, L., Fancher, P.S., and Gillespie, T.D., "An Empirical Model for the Prediction of the Torque Output of Commercial Vehicle Air REFERENCES Brakes," Highway Safety Research Institute, University of Michi gan, Report No. UM-HSRI-78-53, December 1978, 83 p. I. Newcomb, T.P., and Spurr, R.T.,BrakingofRoad Vehicles, Chapman and Hall, Ltd., London, England, 1967,292 p. 2. Limpert, R., "Analysis and Design of Motor Vehicle Brake Sys I 'illS," The University of Michigan, May 1971, 46() p.