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1、Chapter 5 synthesis of planar linkagesThis chapter mainly introduces the characteristics of planar linkage mechanisms, types, and characteristics analysis of four-bar linkages.characteristics and types of planar linkagesfundamental features of four-bar linkagesythesis of four-bar linkage5.1 characte
2、ristics and types of planar linkagesMechanisms are used in a great variety of machines and devicesAll pairs are lower pairscalled lower pair mechanismThe simplest closed-loop linkage is the four-bar linkage four rigid links four lower pairs1. Characteristics of planar linkagesHas a simple structure,
3、 is easy to manufacture, and has a low costHas a large load capacityCan achieve various motions Can transmit motion within a long distanceIt is difficult to perform precision motion2. Types of planar linkagesAD：frameAB,CD：rockerBC：couplerFour bar linkage mechanismABCDside links连架杆: the links connect
4、ed to the frame Crank曲柄：side link can rotate through a full revolutionRocker摇杆：side link cannot rotate through a full revolution, oscillating through an angle coupler连杆：links which are not connected the frame and the driving links Full turning pair整转副: turning pair can have a whole revolutionOscilla
5、ting turning pair摆转副: the pair cannot rotate a full revolutionTypes of the four-bar linkages1.Crank-rocker linkage曲柄摇杆机构2.Double-crank linkage双曲柄机构3.Double-rocker linkage双摇杆机构4.Parallel crank four-bar linkage平行四边形机构5.Isosceles trapezium linkage等腰梯形机构6.Slider-crank linkage曲柄滑块机构7.Rotating guide-bar l
6、inkage转动导杆机构8.Rocking-block linkage曲柄摇块机构9.Sliding guide-bar linkage移动导杆机构10.Rocking guide-bar linkage摆动导杆机构11.Double-slider-crank linkage双滑块机构12.Double rotating block linkage双转块机构13.Sine linkage正弦机构14.Tangent linkage正切机构1(1)crank-rockerlinkage曲柄摇杆机构Link1 can rotate continuously Link3 only oscillate
7、s through an angleCrank-rockerAB：crankCD：rockerDough mixing machine:it consists of a crank-rocker linkageThe coupler curves can be used to generate quite useful path motionos for machine design problemIn this mixer, the coupler curve is like the mixing motion of bandsJaw crusherIt consists of a cran
8、k-rocker linkageThe connecting rod is the moving jaw used to crush the rocksPitchingMechanismforRadarAntenna2(2)Doublecranklinkage双曲柄机构Both link1 and link3 can rotate continuously relative to the frame 1Double-crankAB：crankCD：crankInertia vibrating screen惯性筛机构Drag-link linkageIf the driver AB rotate
9、s clockwise with the constant angular velocity, the slider E makes a slow stroke and returns with a quick strockThe inertia forces can cause vibration of the screen to perform sparation of the materials3(3)Double-rockerlinkage双摇杆机构Both the green link and grey link cannot rotate completely relative t
10、o the blue frame Double-rocker mechanism双摇杆机构AB：rockerCD：rockerCrane used in portsPoint E on coupler can move along a straight line or nearly along a straight lineThe lifted body may move steadily along this pathCraneTipping bucket device汽车自卸翻斗装置retractable undercarriages飞机起落架收放机构(4) Parallel crank
11、four-bar linkage平行四边形机构two opposite links are parallel and equal in lengthAlways have the same angular velocitylinkage mechanism of locomotive wheel机车车轮联动机构lifting mechanism of photography platform摄影平台升降机构opening and closing mechanism of car door汽车车门开闭机构 (5) Isosceles trapezium linkage等腰梯形机构Two rock
12、ers of a double-rocker linkage are equal in lengthTwo equal rockers AB and CD are fixed to the stub axes to form two similar bell-crank leversThis linkage is known as a isosceles trspezium linkageSteeringlinkageofvehicles汽车前轮转向机构(6) Slider-crank linkage曲柄滑块机构Sidelink1isacrankSidelink3isasliderTurnin
13、gpairsAandBarefullturningpairsCisoscillatingturningpairPunching machine压力机(7) Rotating guide-bar linkage转动导杆机构TwolinkscanrotatecompletelyCouplerbecomesasliderinshape刨床主运动机构shaper(8) Rocking-block linkage曲柄摇块机构The crank rotates completelyThe block oscillates about its pivotTipping bucket device汽车自卸翻斗
14、装置Hand pumpLink is made of a cylinder A plunger fixed to the link reciprocates in it(9) Sliding guide-bar linkage移动导杆机构Guidelink4reciprocatesalongtheaxisofthefixedblockSidelink2oscillatesaboutpivotc(10) Rocking guide-bar linkage摆动导杆机构Link 1 can rotates about pivot A completelyLink3 oscillates about
15、the pivot Cmain mechanism of slotting machine插床主机构(11) Double-slider-crank linkage双滑块机构Two side links become sliders which are reciprocating along each axis of cross frameTwo pairs of the same kind are adjacentElliptical trammelThe fixed link 4 is in the form of guides for sliders 1 and 3When two sl
16、iders move along their guide, the link2 will trace an ellipse on a fixed planeThe midpoint of the link2 will trace a circle(12) Double rotating block linkage双转块机构Oldham linkageTwo sliders are rotating completely about their pivots A and B relative to the frame 2Oldhams coupling十字滑块联轴器When the rotati
17、ng blocks of the linkage are replaced by two shaftsThe one acts as a driver and the other as a drivenIt can be used to connect two parallel shafts when the distance between their axes is small(13) Sine linkage正弦机构Called a Scotch-yoke linkageCrank2 rotates about its pivot A completelyLink4 reciprocat
18、es in the fixed link1 to produce a simple harmonic motion压缩机(14) Tangent linkage正切机构Side link 2 oscillates about its pivot ALink 4 reciprocates along its axisLink3 is a slider in shapeRapsons slide linkage3. Evolution and mutation of planar linkageAll kinds of four-bar linkage are relevant to each o
19、ther inherentlyInversion of four-bar linkageConverting a turning pair into a sliding pairExpansion of pin size in aturning pair(1)Inversion of four-bar linkageDiffernent mechanisms can be obtained by fixing different links of a mechanismKnown as inversionRelative motion between links of a four-bar l
20、inkage does not change in different inversionLink1 is a crankLink2 is the couplerLink3 is the rockerLink4 is the frameTurning pairs A and B are full turning pairsPairs C and D are oscillating pairsIf the link1 is fixed instead of link4This makes link2 and link4 rockers and oscillate about their pivo
21、ts A and B respectivelyIt is a double crank mechanismIf the link3 is fixed instead of link4This makes link2 and link4 rockers and oscillate about their pivots D and C, respectivelyDouble-rocker mechanismIf the link1 is fixed instead of link4This makes link2 and link4 cranks and rotate about their pi
22、vots A and B respectivelyLink3 becomes a couplerIt is a rotating guide-bar mechanismIf the link2 is fixed instead of link4, this makes link1 a crankThe block oscillates about its pivots CIt is a rocking-block mechanismIf the link3 is fixed instead of link4The mechanism becomes a sliding guide-bar me
23、chanism(2)Converting a turning pair into a sliding pairIf the radius of the pin D is increased to a length of lCDthe rocker3 is made a slider with a curve This mechanism becomes a curve slider-crank mechanismIf the lCD were made infinite in lengthPoint C would rectilinear motionLink3 could be replac
24、ed by a sliderCalled offset slider-crank mechanismThe path of the slider does not intersect the center axis of crankThe distance from the center axis of the crank to the path of the slider is called an offset, denoted as eIf theoffset e is equal to zeroThe path of slider passes through the crank cen
25、terCalled a inline slider-crank mechanism(3) Expansion of pin size in aturning pairIf we increase the size of the crank pin B until it is larger then the length of the crankThis enlarged crank pin is called an eccentric diskCan be used to replace the crankThe crank consists of a circular disk with c
26、enter B, which is pivoted off-center at A to the frameThe disk rotates inside the ring end of coupler2The motion of the eccentric mechanism is equivalent to that of a slider-crank mechanism having a crank length equal to AB and a coupler of length BC4. Application of the linkagesCan you give some ex
27、amples used in machinery? Crank-rocker: Blender,Crank-slider block: piston engine, punchOthers： bus door driving mechanism， crank-shaper hand-operated well pump 4. Application of the linkages(ct)Please try to type the quite different applications above.They represent three different tasks:1) Path ge
28、neration: concern with the path of a tracer point. For example?2) function generator: the relative motion between links connected to ground is of interest. For example?3) motion generation: the entire motion of the coupler link is of concern. For example?应应用用实实例例：压压力力机机雨雨伞伞generator发发 动动 机机umbrellac
29、ompressor3.曲柄摇块机构曲柄摇块机构(Rock-SliderMechanism)ABC3214摇块摇块功能：功能：Continuous r o t a t i o n连连续续转转动动Reciprocating swing往往复复摆摆动动车车门门开开闭闭机机构构ABC3汽缸汽缸2车门车门14Air compressor空空 气气 压压 缩缩 机机BusdoorDrivingMechanismapplication：Feeding device送送料料装装置置Shifting machine筛筛分分机机R e v o l v i n g p l u n g e r p u m p回回转转
30、柱柱塞塞泵泵早早 期期 的的 飞飞 机机 发发 动动 机机牛牛头头刨刨床床Dump truck自自 卸卸 车车P u m p i n g m a c h i n e抽抽 水水 机机1.Sinusoidalinstrument正弦机正弦机构构AB123AB123从动件从动件3的位移与原动件的位移与原动件1的转角成正比的转角成正比应应 用用实实 例例压压 缩缩 机机缝缝纫纫机机进进针针机机构构2.Doubleslidermechanism双滑块机构双滑块机构AB123AB123AB123(x,y) Ellipsometry椭椭圆圆仪仪3.Double双转块机构双转块机构AB123AB123AB12
31、3应应 用用 实实 例例十十字字滑滑块块联联轴轴器器5.2 fundamental features of four-bar linkagesDistinguishing featuresGrashof criteria（曲柄存在条件 ）Quick-return characteristic（急回特性）Transmission angle（传动角）Dead-point position（死点位置）1.The Grashof condition(曲柄存在条件曲柄存在条件) if the link AB rotate through a full revolution with an angula
32、r velocity clockwise, it must pass through the position AB1 and AB2 From the side length relationship of triangles and ConclusionsLink1 which can rotate through a full revolution must be shortest linkGrashofCriterion:The sum of the length of shortest link and the longest link cannot be greater than
33、the sum of the remaining two linksIf the shortest link is fixed, the linkage acts as a double-crank linkageLinks adjacent to the fixed link will have complete revolutionIf the sum of the length of the shortest and longest links is greater than the sum of the length of the other two linksOnly double-
34、rocker linkage will resultCalled non-Grashof mechanismNo link can rotate through a full revolutionGrashof criteriaA linkage, the sum of the length of the shortest and longest links is less than the sum of the length of the other two links, must have a crankThe crank must be the shortest link1) Let S
35、=length of shortest link L=length of longest link P= length of one remaining link Q= length of other remaining linkThen if S+L=P+Q, the linkage is Grashof and at least one link will capable of making a full revolution with respect to the ground plane. 2) The Grashof condition are independent from th
36、e order of the S,L,P,Q when the mechanism is assembled.Discussion(1)Case 1 S+LP+Q: All inversions will be triple-rocker, no link can fully rotate.Case 3 S+L=P+Q: At the change points, the output behavior will become indeterminate. The common used mechanisms are parallelogram(平行四边形) and antiparallelo
37、gram.Discussion(2)Change points, and indeterminate behavior.How to solve this problem? Table2-3 Type criteria for the revolute four-bar mechanismsLink as the frameS+LP+QS+L=P+QGrashofNon-GrashofChange-pointThe shortest linkDouble-crankDouble-rocker(withoutfullyrotate)orTriple-rockerOpposite to the s
38、hortest linkDouble-rocker(withfullyrotate)Adjacent to the shortest linkCrank-rockerExample：(1)(2)Whicheverlinkistheframe,themechanismisdouble-rockermechanism.IflinkABistheframe.Double-crankmechanismIfLinkCDistheframe.Double-RockermechanismIflinkBCorADistheframe. Crank-rockermechanism(3)SatisfytheGra
39、shofCriterionDouble-crankmechanismParallel-crankmechanismAntiparallel-crankmechanismThelengthoftwocranksarethesame2. Quick return （急回特性）Toggleposition极限位置ThecrankrotatesthroughafullrevolutionwithanangularvelocityanticlockwiseTherockeronlyoscillatesthroughanangleThecrankandcouplerwillformastraightlin
40、eateachextremepositionThe angle between these two limit positions for Crank AB is called acute angle(极位夹角).The crank executes the angle While the rocker oscillates from C1 to C2 through an angle with time t1Average speed of the forward stroke isIf the crank executes the angle while the rocker oscill
41、ates from C2 to C1, through an angle with time t2, the average speed of the return stroke v2 isIt will be observed that the crank angle , corresponding to the forward stroke C1C2 of the rockerIs larger then the return angle, corresponding to the return stroke C2C1The ratio of the average speed v2 of
42、 the return stroke and v1 of the forward stroke is called the coefficient of travel variation, and is denoted as KTo produce a quick-return motionThis ratio must be obviously greater than unitymust be as large as possibleQuick-return mechanismCrank-rocker linkageGuider-bar linkageOffset slider-crank
43、 linkage2.Four-barQuickreturn（急回特性）（急回特性）Definition of the Quick-Return(急回)It is used to describe the fact that the forward stroke is faster than that of the return stroke. The time ratio(TR) defines the degree of quick-return of the linkage. TR= 2 / 1K= f / s = 1 / 2 =180o+ / 180 o - K: travel velo
44、city-ratio coefficient行程速比系数行程速比系数Whendesigningmachinetoolssuchasshapersorpower-drivensawsitisdeisrabletogivetheworkingstrokeofthefolloweraslowerspeedbecauseofthehighresisitancegivethereturnstokeafasterspeedinordertoreduceidletime.1.Offsetslider-crankmechanismOtherquickreturnmechanisms:Ifthedrivingc
45、rankABrotatesclockwisewithconstantangularvelocity,thesliderwilltakealongertimeinitsrightwardstrokethaninitsleftwardstroke.Since =0o,anin-lineslider-crankmechanismhasnoquick-returncharacteristics.2.OscillatingGuide-barMechanismForthislinkagemechanism,happenstobeequalto.IfthecrankABrunscounter-clockwi
46、seataconstantangularvelocity,thentheleftwardstrokeofthefollowerCDistheslowerstrokewhiletherightwardstrokeisthefasterstroke.Reciprocating conveyor往复式运输机Shaper牛头刨床3. Transmission angle & Pressure angle （传动角与压力角）（传动角与压力角）WhatistherelationbetweentheFandFn or (Ft)WhatisthefunctionoftheFn & FtfollowerPres
47、sureangle：Theacuteanglebetweenthedirectionsofthestaticforcetransferredthroughthecouplerandtheabsolutevelocityoftheoutputlinkattheconnectedpoint,suchasCvcF1ABCD234vcFF1F21ABCD234nForce decomposition： active component along the velocity directionactive component perpendicular to the velocity direction
48、Thecouplerisatwo-forcelinkThedirectionofthestaticforceofthecouplerisalongthelineofitspinjointsvcF1ABCD234Pressure angle ForceF2canonlyproducetheaxialcompressionforcetohingesCandD,itshouldbeassmallaspossible.ThegreaterforceF1is,thelargertheeffectivecomponentofpromotemechanismis,andbettertheforcetrans
49、missioneffectis.shouldbeassmallaspossiblen =4050Transmission angle ：1)Defined as the angle between the output link and coupler2)Taken as the absolute value of the acute angle of the pair of angles at the intersection of the two links3)Varies continuously from min.to max. value4)A measure of the qual
50、ity of force transmission at the joint Transmission angle （传动角）（传动角）Transmission anglen= 90- vcFF1F21ABCD234The larger is , the better stress condition is.The smaller is，the better stress condition is.Transmission angle Intheprocessofoperation,transmissionangleisvariedwiththevariationofmechanismposi
51、tioninordertoensuretheforcetransmissioneffect.Transmissionangle：Thecomplementofthepressureangleistransmissionangle.OnlythetangentialcomponentFtcancreatetheoutputtorqueonthedrivenlinkCDTheradialcomponentFrincreasespivotfrictionanddoesnotcontributetotheoutputtorque.Forthisreason,itisdesirablethatisnot
52、toogreatorisnottoosmall.OnlythetangentialcomponentFtcancreatetheoutputtorqueonthedrivenlinkCDTheradialcomponentFrincreasespivotfrictionanddoesnotcontributetotheoutputtorque.Forthisreason,itisdesirablethatisnottoogreatorisnottoosmall.ACBDvBFFvcaAB134Cb2vcABC12F FvB3B123ACTransmission angle Oneextreme
53、valueoccurswhereLinks1and4arecolinearandnonoverlapping;TheotherextremevaluewhereoccursLinks1and4arecolinearandoverlapping;Forsmoothrunningandgoodforcetransmission,keeptheminimumtransmissionangleaboveabout40o.follower3. Transmission angle & Pressure angle （传动角与压力角）（传动角与压力角）4. Toggle positions & Dead-
54、pointsToggleposition极限位置4. Toggle positions & Dead-pointsDead-pointstherockeristhedriverthefollower(crank)iscolinearwiththecouplerthetransmissionangleequalsto0oor180o.theforceappliedtothefollowerpassesthroughthefixedpivotofthefollower.4. Toggle positions & Dead-pointsDead-pointsthemechanismcannotmov
55、ebyatorqueonthedrivingrockerInthissense,thesepositionsarecalleddead-points.Whether does the mechanism have a dead point position depends on which link is the driver.Forthecrank-rockermechanism,whenthecrankisthedriver,thereisnodeadpointpositionbutalimitposition.Whentherockeristhedriver,thereisadeadpo
56、intposition.曲柄摇杆机构，当曲柄为主动件时无死点位置，但有极限位曲柄摇杆机构，当曲柄为主动件时无死点位置，但有极限位置，当摇杆为主动件时有死点位置。置，当摇杆为主动件时有死点位置。For the crank-slider mechanism, when the slider 3 is the driver, there is a dead point position.曲柄滑块机构，当以滑块曲柄滑块机构，当以滑块3 3为主动件时有死点位置。为主动件时有死点位置。For the parallelogram mechanism, when the coupler and the cra
57、nk is collinear, the transmission angle is 0 (a turning point). The driven crank can rotate in two directions, the motion is uncertain, and the parallelogram mechanism may become inverse parallelogram mechanism.平行四边形机构，连杆与曲柄共线时传动角为平行四边形机构，连杆与曲柄共线时传动角为0 （转向点转向点），），从动曲柄可能向正反两个方向转动，机构运动不确定，平行从动曲柄可能向正反两
58、个方向转动，机构运动不确定，平行四边形机构可能变成反平行四边形机构。四边形机构可能变成反平行四边形机构。n For a double rocker mechanism, there is dead point position. nIn actual design, the angle of pendulum is often limited and used to avoid the dead point position.n双摇杆机构，也有死点位置，在双摇杆机构，也有死点位置，在实际设计中常采用限制摆杆的角度实际设计中常采用限制摆杆的角度来避免死点位置。来避免死点位置。Discussion
59、: Pros/cons of the dead-pointsPros: in some circumstances, the dead-points are used to lock the position. For example the clamping device, landing mechanism in airplane.Cons: the dead-points are not good to the transmission. But we can use either the flywheel on the driven crank, or providing the du
60、plicate linkage 900 out of phase to overcome the dead-points.Clamping device夹具AfterasmallforceFappliedonthecouplerhasbeenremovedThereactionforceoftheclampedworkpiecetothelinkABcannotdrivethelinkagetomoveatalllanding mechanism in airplanebrake device of wheelchairSewing machine5. Mechanical advantage
61、Limitpositionanddeadpointpositionareactuallyatthesamelocationofmechanism.Whenthedrivingcomponentiscolinearwiththeconnectingrod,itisthelimitposition.Nearthelimitposition,thevelocityofdrivencomponentisclosetozeroandthemechanicaladvantagewillbeobtained.5. Mechanical advantageMechanical advantage (mA) c
62、an be defined as: mA =Fout / Fin Rivetpullingmachine拉铆钉机拉铆钉机 5.3SynthesisofFour-barLinkagesGraphicalmethod图解法图解法Analyticalmethod解析法解析法Experimentalmethod实验法实验法synthesisofamechanismisthedeterminationofthekinematicdimensionsnecessarytoachievetherequiredmotion.Thebasicprincipleofgraphicalmethod:Construc
63、ttofindallthepivotcenterofrevolutepairsABCDBiCii =1、2、NEiFiFixedhingeA、D：MovablehingeB、C ：thecentreofacircleCircleorcirculararcThekinematicrelation：1.BodyGuidance1.Thefixedpivotshavenotbeendetermined Given:B1C1,B2C2,B3C3，Design:thelinakgemechanismSolve:SincethelocusofthepointBrelativetotheframeisaci
64、rclethecenterofwhichisthefixedpivotA.ThecenterofthecirclepassingthroughthethreepointsB1,B2,B3isthepositionA.Similarly,therevolutecenterDcanbeconstructed.2.ThefixedpivotshavebeendeterminedGiven:E1F1,E2F2,E3F3,A,DSolution:Adeterminedpoint(A)ontheunderminedlink(AB)connectedbyundeterminedmovingrevolutec
65、enter(B)isselectedasacircumferencepoint.Thelocusofthelink(EF)isacircle.Inordertofindthelocusofpoint(A),thelinkEFshouldbestationary.TheconvertingframemethodshouldbeusedsothatthelinkEFisinvertedtobestationary,orimaginaryframe.2.FunctionGenerationGiven:AB1,AB2,AB3,DE1,DE2,DE3Sol:Theconvertingframemetho
66、dshouldbeusedsothatthelinkAB1isinvertedtobestationary,orimaginaryframe.ThepointB2B3arethethreepositionsofpointBrelativetothelinkDE1.TheintersectionofthebisectorsofB1B2andB2B3isthefirstpositionC1.3.DesignofQuickReturnMechanisms(1)Crank-rockermechanismSteps:(a)ChooseDanddrawthetwo(b)limitingpositions,
67、DC1,DC2(b)Cal.(c)ThroughC1constructalineperpendiculartoC1C2.ThroughC2(a)constructalinesothat(b)PC2C1=90-Theintersectionoftwolinesisp.(d)DrawacirclewiththmidpoinofC2PasthecenterandthelengthofthelengthofthelineC2Pasthediameter.(a)(e)ThecircleisthelocusofthefixedpivotA.(2)Offsetslider-crankmechanismKno
68、wn:strokeH,timeratioK,offsete(a)DrawthestrokeH,C1C2(b)Cal.(c)ThroughC1constructalineperpendiculartoC1C2.ThroughC2constructalinesothat PC2C1=90-Theintersectionoftwolinesisp.(d)DrawacirclewiththmidpointofC2PasthecenterandthelengthofthelengthofthelineC2Pasthediameter.(e)Thecircleisthelocusofthefixedpiv
69、otA.(3)Oscillatingguide-barmechanismKnown:K(timeratio)lAD(theframe)Require:lABSolution:Thecrankacuteanglebetweentwolimitingpositionshappenstobeequaltotheangularstrokeoftheguide-barCD.4.PathGenerationDetermine:1、Thelengthofeverylink.2、Thecouplerpoint.The atlas of the coupler curve5.LimitationsofLinka
70、geMechanismsSupposethatthelinkageisusedtoco-ordinatetherotationalangleoftheinputandoutputforfivepositions.Sincethereareonlyfiveunknowsinthissetofequations,thevaluesofthefiveunknowscanbefoundbysolvingtheseequationssimultaneously.Therefore,thislinkagecanco-ordinateexactlyonlyuptofiverelationshipsbetwe
71、entheinputangleandoutputangle.Atotherpositions,therewillbesomeerrors(structuralerror).ProblemsandExercises1.Revolutefour-barLinkageABCD,lBC=100mm,lDC=80mm,lAD=110mm(1)FindtherangeofthevaluesforthelengthlABoflinkABifthelinkageiseither(2)a)acrank-rockermechanismwithcrankABorb)adouble-rockermechanism.(
72、2)Canthelinkagebeadouble-crankmechanismbychoosinglABsuitably?Why?Solution:(1)(a) lABisthecrank.(b)ExcludethecasethateitherlAB or lCD isthecrank.lAB istheordinarylink，lCD isthecrank：80+11090andlAB110orlAB is the longest link，LCDisthecrank：80+lAB100+110 LAB110Soweget：70lAB130lAB isimpossibletobeinfini
73、telong，so：lAB80+100+110=290mmResult:70lAB90or130lAB290Solution:ShowninFig.PracticeOutsideClassBionicinchwormmechanismassignmentdesign and build a mechanism to illustrate the charateristics of four-bar linkagestwo of best students will have the opportunity to give a presentation The larger the angle , the greater the value of K is, the more obvious the quick return can be seen.
