3 ExternallyandInternallyPivotedShoe Brakes TypicalexternallyandinternallypivotedshoebrakesareshowninFigures1 and 2. In all but extremely rare designs, equal forces act upon both shoes to produce equal applied moments about their pivots. External shoe brake control is usually through a lever system that may be driven by electro- mechanical, pneumatic, or hydraulic means. Internal shoe brake control is usually by means of a double-ended cylinder or a symmetrical cam. Calculation of the moments and shoe lengths to achieve a specified braking torque cannot be carried out directly when the two shoes are pivoted as shown in either of these figures and when the opposing shoes are sub- jected to equal moments. The tedious task of manually iterating these formulas to get a satisfactory design under these conditions may be eliminated with the use of computer programs, such as those mentioned in the following sections, that can quickly produce either graphical or numerical design solutions. I. PIVOTED EXTERNAL DRUM BRAKES A. Long Shoe Brakes Externally pivoted, long shoe brakes similar to that shown in Figure 1 are often used as holding brakes. As its name implies, a holding brake is to hold a shaft stationary until the brake is released. The compression spring on the left- hand side of the brake in Figure 1 applies a clamping force to the brake shoes Copyright © 2004 Marcel Dekker, Inc. oneithersideofthebrakedrumtoholditstationarywithouttheneedfor externalpower.Electricalcurrentthroughthesolenoidontheleftsideofthe assemblyreleasesthebrakeandholdsitopenforaslongasvoltageisapplied tothesolenoid.Otherholdingbrakesmayuseslightlydifferentmechanical arrangementsandmayuseeitherahydraulicorapneumaticcylinderto releasethebrake. Oneoftheapplicationsofaholdingbrakeisinthedesignofanoverhead crane.Thevalueofaholdingbrakeisthatitallowsaloadthathasbeenraised tobeheldinpositionwithoutexternalpower.Thisisalsoasafetyfeature becausethecranewillnotallowitsloadtoberaisedorlowereduntilthebrake isintentionallyreleased.Likewise,thesebrakesarealsousedinhoists,in punchandformingpresses,andinsomeconveyorsystems,forsimilar reasons. Webeginthederivationofthegoverningequationsforthebraking torquebyconsideringonlyoneshoe(Figure3)andthenextendingthose resultstomorethanoneshoe.Undertheassumptionthattheshoe,lever,and drumareallrigid,andthatthestress–strainrelationsoftheliningarelinear, F IGURE 1Externallypivotedshoebrake.(CourtesyAutomation&ProcessTech- nology Div., Ametek Paoli, PA.) Chapter 332 Copyright © 2004 Marcel Dekker, Inc. F IGURE 2 Internal pivoted shoe brake. (Courtesy of Dyneer Mercury Products, Canton, Ohio.) Externally and Internally Pivoted Shoe Brakes 33 Copyright © 2004 Marcel Dekker, Inc. the pressure p at any position along the lining due to infinitesimal rotation yh of the shoe about pivot A will be given by p ¼ kRyh sin B ð1-1Þ in terms of the quantities shown in Figure 3. Upon the introduction of the maximum pressure, written as p max ¼ kRyh ðsin fÞ max ð1-2Þ we find that kRyh ¼ p max sin fðÞ max ð1-3Þ so that after substitution from equation (1-3) into equation (1-1), we find that the pressure may be written in terms of the maximum pressure as p ¼ p max sin fðÞ max sin f ð1-4Þ F IGURE 3 Geometry involved in calculating moment M p about pivot point A. Chapter 334 Copyright © 2004 Marcel Dekker, Inc. Withthisexpressionforpressureasafunctionofposition,thetorqueonthe drumwillbetheintegralovertheshoelengthoftheincrementalfrictionforce A(prwÀdf)actingonthesurfaceofadrumofradiusr.Thus T¼Ar 2 w p max sinfðÞ max Z f 2 f 1 sinfdfð1-5Þ inwhich(sinf) max denotesthemaximumvalueofsinfwithintherange f 1 VfVf 2 .Integrationofequation(1-5)yields T¼ Ap max r 2 w sinfðÞ max cosf 1 Àcosf 2 ðÞð1-6Þ inwhichf 1 istheanglefromradiusRbetweenthedrumaxisandpivotA tothenearedgeofthedrumsectorsubtendedbythebrakelining.As drawninFigure3,anglef 2 ismeasuredfromradiusRtowardthefaredge ofthebrakelining.Hencetheanglesubtendedbytheshoeisgivenby f 0 ¼f 2 Àf 1 ð1-7Þ TocalculatethemomentthatmustbeappliedaboutpivotAinFigure3 toobtainthetorquefoundbyequation(1-6),wefirstcalculatethemoment reactionatthepivotduetoboththeincrementalnormalforcesandthe incrementalfrictionforcesactingonthelining.Anequalandopposite momentmust,ofcourse,besuppliedtoactivatethebrake. Radialforceprwdfoneachincrementalareaalsocontributestoa pressuremomentM p aboutpivotA.RelativetothegeometryinFigure3,and withtheaidofequation(1-4),thismomentmaybewrittenas M p ¼ Z f 2 f 2 pwrdfðÞRsinf¼ p max wrR sinfðÞ max Z f 2 f 1 sin 2 fdfð1-8Þ whichintegratesto M p ¼ p max wrR 4sinfðÞ max 2f 0 Àsin2f 2 þsin2f 1 ðÞð1-9Þ wheref 0 isgivenbyequation(1-7).Thismomentispositiveinthecounter- clockwisedirection,anditsalgebraicsignisindependentofthedirectionof drumrotationrelativetothebrakelever’spivotpoint. ReactivemomentM f atpivotAduetothefrictionforceactingonthe shoemaybecalculatedusingthegeometrysketchedinFigure4.Thus, M f ¼ Z f 2 f 1 Apwr dfðÞR cos f À rðÞ ð1-10Þ ¼ Ap max wr sin fðÞ max Z f 2 f 1 R cos f sin f À r sin fðÞdf Externally and Internally Pivoted Shoe Brakes 35 Copyright © 2004 Marcel Dekker, Inc. F IGURE 4 Geometry involved in calculating moment M f about pivot point A. Copyright © 2004 Marcel Dekker, Inc. Integrationofequation(1-10)yields M f ¼ Ap max wr 4sinfðÞ max Rcos2f 1 Àcos2f 2 ðÞÀ4rcosf 1 Àcosf 2 ðÞ½ð1-11Þ Herethequantityenclosedbythesquarebracketsdeterminesthealgebraic signofM f andmaycauseittobezero.Thephysicalsignificanceoftheal- gebraicsignfornonzerovaluesofmomentM f dependsuponthedirection ofrotationNofthedrum. Iftherotationistowardthepivot,asinFigure4,apositivevalueofM f signifies a clockwise moment about the pivot that applies the brake by forcing the shoe against the drum, which would cause self-locking. Therefore, a negative or zero value for M f from equation (1-11) is required to produce either a counterclockwise or a zero moment, respectively, about the pivot point. The interpretation is reversed if the drum rotation N is away from the pivot. In this case a positive value from equation (3.11) indicates a counter- clockwise rotation of the shoe about the pivot that tends to release the brake. Obviously, a negative value in this situation indicates a clockwise moment about the pivot that tends to rotate the shoe toward the drum. From these observations it follows that brake activation requires an applied moment M e about the pivot point A such that M p þ M f ¼ M e > 0 N away from the pivot M p À M f ¼ M e > 0 N toward the pivot ð1-12Þ where M f itself, as calculated from equation (1-11), must be negative or zero when rotation N is toward the pivot and positive or zero when it is away from the pivot—hence the minus sign in the second of equations (1-12). Self-locking is of use only when the brake is to serve as a backstop or as an emergency brake during control failure. Otherwise, self-locking is gen- erally to be avoided because it does not allow the braking torque to be controlled by the control of M e . B. Short Shoe Brakes Short shoe brakes are generally defined as those for which the angular dimension of the brake, f 0 , is small enough (generally less than 20j) that sin f g (sin f) max and p g p max so that with these restrictions equation (1-5) may be approximated by T ¼ Apwr 2 f 0 ¼ ArF ð1-13Þ where F ¼ pwr f 0 ð1-14Þ Externally and Internally Pivoted Shoe Brakes 37 Copyright © 2004 Marcel Dekker, Inc. istheforceexertedontheshortshoe.Applicationoftheseapproximationsto equation(1-9)beforeintegrationyields M f ¼AFRcosf 1 ÀrðÞð1-15Þ Similarly,applicationoftheseapproximationstoequation(1-10)before integrationyields M p ¼FRsinf 1 ð1-16Þ sothatsubstitutionintoequation(1-12)withtheminussignineffectreveals thattheshortshoewillnotbeself-lockingif sinf 1 ÀAcosf 1 À r R  >0ð1-17Þ II.PIVOTEDINTERNALDRUMBRAKES TheequationsderivedinSectionIAdealingwithlongexternalshoebrakes applyequallywelltointernalshoedrumbrakes.Thereisoneessential difference,however,thatdoesnotappearexplicitlyintheequationsthem- selves:ThephysicalsignificanceofpositivevaluesofmomentsM p andM f is different.Thegeometryusedtoobtaintheserelationsforinternalshoebrakes isshowninFigures5and6;thedifferentinterpretationsforthevarious combinationsofdirectionofrotationandinternalorexternalshoesarelisted inTable1.Inthattablerotationofthedrumfromthefarendoftheshoes totheendnearthepivot(termedrotationfromthetoeofthebraketotheheel) isindicatedbyanarrowpointingtowardtheletterp;rotationintheopposite directionisindicatedbyanarrowpointingawayfromtheletterp.Theacro- nymcwindicatesclockwiserotation(orthedirectionofrotationofanad- vancingright-handscrew),andccwindicatescounter-clockwiserotation. FromFigure5itfollowsthat dM f ¼ Awrp d fðÞr À R cos fðÞ ð2-1Þ This is the negative of the integrand in equation (1-10). The rotation indicated causes the shoe to pivot in the counterclockwise direction about A; but because equation (1-10) used the negative of the integrand above, the rotation shown corresponds to a negative M f value as calculated using either equation (1-10) or equation (1-11). Hence, negative M f from these formulas implies counterclockwise rotation and positive M f corresponds to clockwise rotation of the shoe about its pivot. Braking requires a moment M a applied to the shoe as given by M p À M f ¼ M a N away from the pivot M p þ M f ¼ M a N toward the pivot Chapter 338 Copyright © 2004 Marcel Dekker, Inc. forinternalshoes.Thephysicalsignificanceofthealgebraicsignsassociated withthemomentexpressionsderivedintheprecedingsectionsasappliedto externalandinternalbrakesisdisplayedinTable1.Itismaybehelpfulthe rewrite the equations for either internal or external brakes in terms of different symbols if the use of a single set of equations for two different cases becomes too confusing. After using these equations enough to become familiar with them, the reader may find that analysis is easier if they are again combined into a single set, as has been done here. Drum brake efficiency may be measured in terms of the ratio of the torque produced by the brake itself to the torque required to activate the brake, also known as the shoe factor; namely, T M a ¼ T M p F M f ð2-2Þ Brake efficiency is generally not a design factor in the analysis of drum brakes because it is dependent on too many factors [f 1 , f 2 , r/R, A, w, and (sin f) max ] F IGURE 5 Geometry for calculating the moment due to friction about point A for an internal shoe brake. Externally and Internally Pivoted Shoe Brakes 39 Copyright © 2004 Marcel Dekker, Inc. to make it useful. More significance is usually associated with brake life, heat dissipation, fading, and braking torque capability. III. DESIGN OF DUAL-ANCHOR TWIN-SHOE DRUM BRAKES For both external and internal shoes and for either direction of rotation a positive M e value indicates that an external moment of that magnitude must be applied to activate the brake. The formulas also clearly indicate that the extent of the braking action may be controlled by controlling this activation moment. The role of M f , the moment due to friction, in determining the required activation moment M e may be seen by returning to equation (1-11) F IGURE 6 Geometry for calculating the moment due to pressure about point A for an internal shoe brake. Chapter 340 Copyright © 2004 Marcel Dekker, Inc. [...]... ELECTRIC BRAKES Common usage has associated the term electric brakes with friction brakes which are electrically activated, rather than with those brakes that rely upon electrical and magnetic forces rather than friction to provide the braking torque Typical electric brakes are pictured in Figures 18 and 20 Both are single-anchor drum brakes that use the servo action associated with these brakes to... hold for internal brakes except that the directions of rotation are reversed for the same algebraic signs Since most brakes are designed for rotation in both directions, it is generally convenient to combine these criteria into a single criterion, which is that self-locking of both internal and external drum brakes may be avoided if Mf À1 V Vþ1 ð3-3Þ Mp Selection of shoe and drum angles and dimensions... two brakes which emphasize their means of operation are given in Figure 21 These brakes were designed for use with highway trailers where a quick response time may be important They have both fewer total parts and fewer exposed parts than either hydraulic or air brakes, but do not have as great a braking torque for a given size of drum and shoes Copyright © 2004 Marcel Dekker, Inc Externally and Internally... use because they allow the design engineer to compare the effects of different realistic pressure distribution and to design drums and shoes whose rigidity will induce particular pressure distributions over the primary and secondary shoes The first of the two pressure distributions considered is a synthesis of (1) the sinusoidal distribution associated with a nondeforming drum and shoe, generally associated... the ratio Mf /Mp lie between À1 and +1 is equivalent to À 1 Mf 1 V V A A AMp ð3-4Þ Since pmax, (sin f)max, and A cancel out when equation (1-11) is divided by the product of A and equation (1-9), the ratio Mf/(AMp) is a function of only three quantities: r/R, f1, and f2 Thus, Mp/(AMp) may be plotted as a function of f2 for fixed values of r/R and f1, as in Figures 7 and 8 Criterion (3.4) also can be... drum brakes and 1/A < 0 pertains to internal drum brakes, so these values may be shown on the left-hand ordinate of these graphs by relating them to the limiting values of Mf/(AMp) according to relation (3.4), namely, that at the lower limit, À1=A ¼ Mf =ðAMp Þ Copyright © 2004 Marcel Dekker, Inc 44 Chapter 3 and that at the upper limit, 1=A ¼ Mf =ðAMp Þ Consequently, the ordinates on the right-hand... sin f1 REFERENCES 1 Burr, A H (1981) Mechanical Analysis and Design New York: Elsevier 2 Juvinal, R C (1983) Fundamentals of Machine Component Design New York: Wiley 3 Fazekas, G A G (1958) Some basic properties of shoe brakes Journal of Applied Mechanics 25:7–10 4 Orthwein, W C (1985) Estimating torque and lining pressure for bendix type drum brakes SAE Paper 841234, SAE Transactions 86:5.617–5.622... construction facilitates the design of a self-adjusting mechanism for automotive use, it does not entirely eliminate the difference in wear between the two shoes, and it introduces additional labor to calculate brake torque and lining pressure A FIGURE 12 Schematic of a Bendix, or servo, single-anchor brake Copyright © 2004 Marcel Dekker, Inc Externally and Internally Pivoted Shoe Brakes 51 program to ease... latter tasks is described and demonstrated in the following paragraphs With this program it is easy to show that relatively small changes in the pressure distribution along either shoe may produce large changes in the braking torque Although calculation of the braking torque and consideration of the design of brakes of this type appears to be omitted from most of the machine design texts now in print,... anchor pin (Figures 12 and 13), are the same as those given derived in Section 1, which are that the moments (positive in the clockwise FIGURE 13 Primary link force and incremental pressure and friction forces Copyright © 2004 Marcel Dekker, Inc 52 Chapter 3 direction) about the pivot due to the pressure and the frictional forces are given by Z f2 Mp ¼ rwR p sin f df ð5-1Þ f1 and Z Mf ¼ Awr f2 f1 pðr . ExternallyandInternallyPivotedShoe Brakes TypicalexternallyandinternallypivotedshoebrakesareshowninFigures1 and 2. In all but extremely rare designs, equal. dissipation, fading, and braking torque capability. III. DESIGN OF DUAL-ANCHOR TWIN-SHOE DRUM BRAKES For both external and internal shoes and for either direction