Tài liệu Clutches and brakes design and selection P9 ppt

9 Centrifugal,One-Way,andDetent Clutches Thesearespecial-purposeclutchesthatareusedinautomatictransmissions, indevicesforbringinghigh-speedmachineryuptospeed,inchainsaws,in conveyordrives,andinsimilarindustrial,vehicular,andlarge-andsmall- equipmentapplications.Thecentrifugalclutchesprovideaspeed-dependent torquewhichactsonlywhentherotationalspeedexceedsaparticularvalue; theone-way,oroverrunning,clutchesprovideatorquethatisnotspeedde- pendentoncetheyareengaged,butisdependentonthedirectionofrota- tion;andthedetentclutchesprovideatorquethatcannotexceedaprescribed value. I.CENTRIFUGALCLUTCHES Acentrifugalclutchmaybedescribedasconsistingofaninnercylinderthatis attachedtotheinputshaftandanouterhousingthatisattachedtotheoutput shaft,asinFigure1.Sectorsoftheinnercylinderarecutouttoallowittobe fitted with weights that can slide radially outward as the inner cylinder rotates so that the weights are forced against the outer housing by centrifugal force and thereby transmit torque to the outer housing. Centrifugal clutches de- signed for lower power transfer may use simpler designs. In some chain saws, for example, it is the weights themselves that are recessed to accept radial guides from the central shaft. Copyright © 2004 Marcel Dekker, Inc. Because of the variety of centrifugal clutch designs, their analysis will be described in general terms. Let A denote the cross-sectional area of each weight in a plane perpendicular to the axis of rotation, written in the form of an annular sector of angle f o as A ¼ cf o r 2 o ð1 À h 2 Þð1-1Þ where h = r i /r o . Parameters h and c are factors that may be used to express other cross-sectional areas in this form of equation (1-1). When h =0,c = 1/2, and f o =2k, area A in equation (1-1) becomes that of a disc of radius r o . Let w denote the width of each weight, measured in a direction parallel to the axis of rotation, and let g represent the mass density of the weights. If F IGURE 1 Centrifugal clutch. (Courtesy Dana Corp., Inc., Toledo, OH.) Chapter 9192 Copyright © 2004 Marcel Dekker, Inc. thestaticdeflectionofaretainingspringattachedtoeachmassisy s ,thenits springconstantkisgivenby k¼gA wg y s ð1-2Þ inwhichgistheaccelerationofgravity,takentobe9.8067m/sec 2 ,or32.2ft/ sec 2 . Denotetheradiustothecenterofgravityofeachweightbyr c .Thenthe centrifugalforceactingoneachweightasitrotatesatangularvelocity x abouttheaxisoftheclutchandmovesoutwardadistanceyisthengivenby F¼gwAðr c þyÞ x 2 Àgn yþy s y s ð1-3Þ wherethespringconstantmaybeincreasedbythefactorntoholdeachweight moresecurelyagainstitsstopatlowrotationalspeeds. Consideraprototypeweightasbeingmadefromasectorofathick cylinderwhoseinnerradiusisr i andwhoseouterradiusisr o .Formthesector bycuttingthecylindertolengthw,whichwillbethewidthofthesector,and thencutthecylinderwithtworadialplanesseparatedbyanglef o .Retainone ofthetwosectorsthatsubtendanglef o astheprototypeweightshowninlater Figure3(a).Theradiusofgyrationofthisweightabouttheaxisoftheoriginal cylinder is given by r c ¼ r o ffiffiffi 2 p ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 À h 2 p ð1-4Þ In order to express the radius of gyration of other geometries in this form, let r c ¼ Er o ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 À h 2 p ð1-5Þ The torque that can be delivered by N of these weights after they have moved outward a distance y to make contact with the inner surface of the housing at radius r o may be written as T ¼ Ar o F ¼ Agwr o NA ðr c þ yÞ x 2 À gnð1 þ DÞ Âà ð1-6Þ where D = y/y s . This relation may be solved for the w required for the clutch to transmit torque T at angular speed x to get w ¼ T NAgf o cr 3 o ð1 À h 2 Þ ðr c þ yÞ x 2 À gnð1 þ DÞ ! ð1-7Þ Centrifugal, One-Way, and Detent Clutches 193 Copyright © 2004 Marcel Dekker, Inc. Maximumpressureontheliningmaybefoundfrom F¼r o w Z f o =2 Àf o =2 pcosf x ¼r o wp max Z f o =2 Àf o =2 cosðfÞ 2 df ¼ p max 2 r o wðf o þsinf o Þ ð1-8Þ uponusingthepressuredistributionfromequation(1-2)inChapter4.Hence, p max ¼ 2F r o wðf o þsinf o Þ ð1-9Þ Theangularvelocityoftheinputshaftwhentheweightsmakeinitial contactwiththedrummaybefoundbysettingthesquarebracketinequation (1-6)equaltozero.Substitutionof x =2kn/60,wherenisinrpm,followedby solvingtheresultingexpressionforn,yields n¼ k 30 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n g r c ðhÞþy ð1þDÞ r ð1-10Þ TheroleofparametershandEinequation(1-5)uponforceF,equation (1-3),andthereforeuponp max ,equation(1-9)andthespeedatwhichtheyfirst contactthedrumareshowninFigures2(a)through(d). Observethatthevariationofthepressure,andhencetheforce,thateach weightexertsagainstthedrumisalinearfunctionofparameterEandthatit becomesanearlylinearfunctionofh,andhenceofr i ,forhgreaterthanabout 0.3.Thedependenceofthewidthofeachweight,however,becomesincreas- inglynonlinearashincreasesandasEdecreases.Therotationalspeedfor initialcontactisalsonearlylinearforh<0.6,especiallyforthelargervalues ofE. Example Designacentrifugalclutchtoprovideatorqueof2400N-mwhentherota- tionalspeedreaches870rpmusingsectorweightshavingthegeometryshown inFigure3(a).Preferredcharacteristicsarethatinitialcontactbetween weightsanddrumoccuratbetween220and230rpmandthatthewidthof theweightsbelessthan30cm.Assumealiningcoefficientoffrictionof0.35 anddesign foraninsidedrumradius(minustheliningthickness)of15cm,a displacementyof3mmforthe segments to contact the drum, and a static deflection of 1 mm. The segments are to be made from an iron alloy having a nominal density of 7880 kg/m 3 , and a safety factor of 3.5 is mandated. Hold Chapter 9194 Copyright © 2004 Marcel Dekker, Inc. F IGURE 2 (a) Variation of pressure (kPa) with h for E = 0.2, 0.4, 0.6, and 0.8 for curves 1, 2, 3, and 4, respectively. (b) Variation of width (cm) with h for E = 0.2, 0.4, 0.6, and 0.8 for curves 1, 2, 3, and 4, respectively. (c) Variation of pressure (kPa) with E for h = 0.2, 0.4, 0.6, and 0.8, respectively. (d) Variation of contact speed (rpm) with E for h = 0.2, 0.4, 0.6, and 0.8 for curves 1, 2, 3, and 4, respectively. Centrifugal, One-Way, and Detent Clutches 195 Copyright © 2004 Marcel Dekker, Inc. F IGURE 3 (a) Sector cross section. (b) Curve 1: pressure P (kPa) vs. h; Curve 2: initial contact speed n (rpm) vs. h. (c) Sector width w (cm) vs. h. Chapter 9196 Copyright © 2004 Marcel Dekker, Inc. theweightsagainsttheirrestpositionwithaforce1.2timestheirweight. Maximumliningpressureoflessthan440kPaispreferred. Becausetheliningpressureonasegmentdecreaseswithangleufromthe centerlineofthatsegmentaccordingtoequation(4-2),usesixweightstogeta greaterforcetransfer,eachsubtendingananglef n =42j. Beginthedesignprocessbyplottingpressurep,contactspeedn,and widthwagainsthbysubstitutingthefollowingvaluesintoequations(1-1) through(1-3),(1-5)through(1-7),and(1-9). T¼24;000N-mr o ¼150mg¼7880kg=m 3 n¼870rpmc¼0:50 y¼0:003my s ¼0:001mn¼1:2f o ¼42jA¼0:35 D¼3N¼6E¼ 1 ffiffiffi 2 p g¼9:8067m=sec 2 TheseplotsareshowninFigure3(b)and(c). Figure3(b)showsthataninitialcontactspeedbetween220and230rpm maybehadforhbetween0.6and0.7,Figure3(c)showsthatthecorrespond- ingwidthofthesectorwouldbelessthan30cm.Substitutingh=0.65into equation(1-10)yieldsn=226.59rpm,whichiswithinthedesiredrange.This iscloseenoughtothepreferredvalueof225rpmformanualiterationofhto findthat n¼225:001rpmath¼0:6367 Thewidthofeachweightandthemaximumliningpressurecorrespondingto h=0.6357arefoundtobe w¼23:8cmandp max ¼304kPa bysubstitutionintoequations(1-7)and(1-9),respectively.Therequired springconstantmaybefoundbysubstitutingfromequation(1-1)into equation(1-2)toget k¼wcf o r 2 o g g y s nð1Àh 2 Þð1-11Þ Substitutionintothisexpressionyields k¼1366N=mm II.ONE-WAYCLUTCH:THESPRINGCLUTCH Wiebuschgavethefirstdescriptionofthisclutch,showninFigure4,in1930 [1]. As may be soon from the figure, it consists of a helical spring snugly, wrapped about both the input and output hubs, parts 1 and 3 in Figure 4, but is attached to neither of them. If the input hub tends to turn in the direction that causes the helix to tighten, the increased friction between the spring and hubs tends to resists any further relative rotation. Relative rotation in the Centrifugal, One-Way, and Detent Clutches 197 Copyright © 2004 Marcel Dekker, Inc. other direction, however, tends to loosen the helix, and relative rotation may proceed with only a relatively small restraint by the spring clutch. Although Wahl [2] appears to have derived a more accurate expression for the torque that may be transmitted, T t , agreement between the Wiebusch theory and experiment seems to be close enough to justify use of the simpler relationship, which is T t ¼ Elr h 1 R 2 À 1 R 1  e 2kNA À 1 ÀÁ ð2-1Þ in terms of the elastic modulus E of the spring material, the moment of area I of the spring wire in bending, the radius R 1 of the neutral surface of the wire in helix 4 in Figure 4 when it is free of external load, the radius R 2 of the wire when the helix is in tight contact with hubs 1 and 5 in the figure, and the number of turns N on one hub if both hubs have the same number of turns. If both hubs do not have the same number of turns, N is the smaller of the two. The friction coefficient is represented by A, and r h denotes the hub radius. Wiebusch found that the torque T u in the unwinding direction was approximately equal to T u ¼ Elr h 1 R 2 À 1 R 1  1 À e À2kNA ÀÁ ð2-2Þ F IGURE 4 Spring clutch and its components. (Courtesy Warner Electric Brake & Clutch Co., South Beloit, IL.) Chapter 9198 Copyright © 2004 Marcel Dekker, Inc. Equation(2-1)obviouslyholdsforatorquelessthanthatwhichcor- respondstothemaximumforcethancanbecarriedbythespringwireatyield. KaplanandMarshall[3]haveindirectlysuggestedthatthelimitingtorque satisfiestheinequality T max Ðbt 2 1:05 2r h R 1 ÀR 2 À t 2  ð2-3Þ forrectangularwirewhosedimensionintheradialdirectionistandwhose dimensionintheaxialdirectionofthehelixisb. III.OVERRUNNINGCLUTCHES:THEROLLERCLUTCH TheseclutchesaredesignedtotransmittorquefromshaftAtoshaftBwhen shaftAtendstorotatefasterthanshaftBbuttodisengagewhenshaftB rotatesfasterthanA.Detailsoffourdesignsthataccomplishthisareshownin Figure5,whichshowsthattheclutchconsistsoftwoconcentricraces,in which one is circular and the other consists of a series of cams, with a roller under, or above, each cam. Relative rotation which wedges the rollers be- tween the narrow portion of the cam and the circular surface of the other race forces both races to rotate together, while relative rotation in the opposite direction frees the rollers and allows the two races to rotate at different an- gular rates. In particular, if the cams are cut in the outer race and tapered in the direction shown in Figure 5(a), (b), and (c), rotation of the inner race in the clockwise direction will cause the rollers to wedge themselves between the two races so that the outer race must also rotate in the clockwise direction, that is, when x i > x o If the outer race is then accelerated to a rotational speed greater than that of the inner race so that x o > x i the roller will move to the larger ends of the cam and the outer race is free to accelerate to a speed greater than that of the inner race. The sequence just described is, for example, that used in starting gas turbines with an electric motor to get them up to operating speed, at which point the turbine accel- erates under its own power and disengages the starter motor, which is then shut off. If the cam surface is cut in the inner race and tapered as shown in Figure 5 (d), clockwise rotation of the outer race will drive the inner race whenever x o > x i Centrifugal, One-Way, and Detent Clutches 199 Copyright © 2004 Marcel Dekker, Inc. Accelerationoftheinnerraceintheclockwisedirectionwillcausetheclutch todisengagewhenever x o < x i asisobviousfromthetapergeometry.Theseclutchesaresaidtobefree- wheelingoroverrunningwhentherelativerotationoftheraceissuchthatno torqueistransmittedfromonetotheother. FromthegeometryofFigures5itfollowsthatthetorquetransmittedto a roller and a convex race is limited by the maximum contact stress that can be sustained along the line of contact (actually, a narrow strip after the surfaces have deformed slightly) between the roller and the race with the smaller radius of curvature. j xx ¼ À2F k 2 a ða 2 þ 2x 2 þ 2z 2 Þ z a C À 2k z a À 3xzA þ Að2x 2 À 2a 2 À 3z 2 ÞA þ 2Ak x a þ 2Aða 2 À x 2 À z 2 Þ x a C ð3-1aÞ j zz ¼À 2F k 2 a zðaC À xA þ AzAÞð3-1bÞ j xz ¼À 2F k 2 a z 2 A þ Aða 2 þ 2x 2 þ 2z 2 Þ z a C À 2kA z a À 3AxzA hi ð3-1cÞ away from the contacting surfaces and by j xx ¼ÀA 4F ka x a À x 2 a 2 À 1  1=2 "# ; x ð a ¼À 2F ka 1 À x 2 a 2  1=2 þ 2A x a "# ; Àa V x V a ð3-2aÞ j xx ¼ÀA 4F ka x a À x 2 a 2 À 1  1=2 "# ; x Ð À a If a finite element analysis program with contact stress capability is not available, the pertinent stress components may be estimated from an analysis by Smith and Liu [4] for the contact (Hertzian) stresses between two parallel Chapter 9200 Copyright © 2004 Marcel Dekker, Inc. [...]... 1 and 2 30j 130j between detents 2 and 3 between detents 8 and 3 and that the angular separation between detent centers 1 and 5 is given by 25j between detents 1 and 2 30j between detents 2 and 3 35j between detents 3 and 4 40j between detents 4 and 5 130j between detents 1 and 5 The second instance occurs when balls 8 and 1 engage detents 1 and 5, where ball centers 8 and 1 are separated by 75j and. .. clutch Ball and detent arrangements in clutches where indexing (maintaining a constant angular relation between input and output shafts) after an overload is not required usually are arranged in axial symmetry in order to reduce shaft vibration and noise Copyright © 2004 Marcel Dekker, Inc Centrifugal, One-Way, and Detent Clutches 221 Many manufacturers of ball and detent overload clutches designed for... and detent centers 3 and 4 are Copyright © 2004 Marcel Dekker, Inc Centrifugal, One-Way, and Detent Clutches 223 separated by 35j and 4 and 5 are separated by 40j for a combined separation of 75j The third and last instance is when balls 7 and 1 engage detents 4 and 7 Placing ball-detent pairs at different radii eliminates engagement except at the index position, but those ball and detent locations... radii, and Young’s moduli of the components in contact, i.e., a roller and the outer race or a roller and the inner race Since the trios of quantities v1, r1, E1 and v2, r2, E2 enter symmetrically into the expression for a, either trio may be associated with a roller and the other trio associated with the inner race Coordinates x and z lie in the circumferential and radial directions, respectively, and. .. ball -and- detent pairs are engaged before the plates reindex The first instance occurs when balls 8 and 3 engage detents 1 and 5 This is because the angular separation between detents 8 and 3 is the same as that between detents 1 and 5 In particular, from the preceding table of the angular positions of the ball -and- detent pairs and the angular separation between centers we find 75j 25j between detents 8 and. .. considered here FIGURE 18 Forces acting on a single wedge tooth (Courtesy Horton Mfg Co., Minneapolis, MN, and Machine Design, Penton Press, Cleveland, OH.) Copyright © 2004 Marcel Dekker, Inc Centrifugal, One-Way, and Detent Clutches 219 FIGURE 19 Overload release clutch using clutch plates and wedge-shaped release cam (Courtesy of Carlyle Johnson Machine Co., Manchester, CT.) The first is shown in... OVERRUNNING CLUTCHES: THE SPRAG CLUTCH A representative sprag clutch is shown in Figure 8 These clutches are also direction depenent, but they differ from the roller clutches in that sprags are used rather than circular cylindrical rollers Sprags are cylinders whose cross section, as shown in Figure 9, is designed to allow them (1) to engage and Copyright © 2004 Marcel Dekker, Inc Centrifugal, One-Way, and. .. Figure 14 These values of rs and ri may be substituted into equations (3-1) and (3-2) and values of rs and ro substituted into equation (3-3) or into a finite element program for contact stresses at the inner and outer radii, to find the minimum radius of curvature rs, shown in Figure 14, that will give a permissible stress for these sprag profiles for the inner race (IR) and for the outer races (OR)... stresses at the IR and OR for the sprag configurations shown in Figure 11 Sprag overrunning clutches have speed envelopes within which they can operate as designed Although these envelopes have the same general shape, the nature of the envelope in the third quadrant (that to the left of the vertical axis and below the horizontal axis) may vary as shown in Figures 15 and 16 for sprag clutches Recommended... Centrifugal, One-Way, and Detent Clutches 217 excess of 1200 rpm in the first and third quadrants, the IR rotational speed tends to be only 50 rpm less than the OR speed V TORQUE LIMITING CLUTCH: TOOTH AND DETENT TYPES Although tooth clutches, as pictured in Figure 17, are usually used for positioning one shaft relative to the other, they may, in an emergency, also serve as overload detent clutches, because . inFigure3(a).Preferredcharacteristicsarethatinitialcontactbetween weightsanddrumoccuratbetween22 0and2 30rpmandthatthewidthof theweightsbelessthan30cm.Assumealiningcoefficientoffrictionof0.35 anddesign foraninsidedrumradius(minustheliningthickness)of15cm,a. indevicesforbringinghigh-speedmachineryuptospeed,inchainsaws,in conveyordrives,andinsimilarindustrial,vehicular,andlarge-andsmall- equipmentapplications.Thecentrifugalclutchesprovideaspeed-dependent