http:/pic.sagepub.com/Engineering ScienceEngineers, Part C: Journal of Mechanical Proceedings of the Institution of Mechanical http:/pic.sagepub.com/content/217/10/1117The online version of this article can be found at: DOI: 10.1243/0954406033225171351117 2003 217:Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering ScienceL-I WuCalculating conjugate cam profiles by vector equations Published by: http:/www.sagepublications.comOn behalf of: Institution of Mechanical Engineers can be found at:Engineering ScienceProceedings of the Institution of Mechanical Engineers, Part C: Journal of MechanicalAdditional services and information for http:/pic.sagepub.com/cgi/alertsEmail Alerts: http:/pic.sagepub.com/subscriptionsSubscriptions: http:/www.sagepub.com/journalsReprints.navReprints: http:/www.sagepub.com/journalsPermissions.navPermissions: http:/pic.sagepub.com/content/217/10/1117.refs.htmlCitations: What is This? - Oct 1, 2003Version of Record at Shanghai Jiaotong University on January 13, 2013pic.sagepub.comDownloaded from Calculating conjugate cam pro? les by vector equationsL-I WuDepartment of Power Mechanical Engineering, National Tsing Hua University, Hsinchu 30013, TaiwanAbstract:This paper presents an analytical approach for determining the pro? les of conjugate disccams. For a conjugate cam mechanism, its two normal lines through the points of contact and the lineof centres must always intersect at a common point, which is an instant centre. On this basis, thecontact points between the conjugate cam and the follower can be determined according to thelocations of instant centres and follower position. The cam pro? les, the paths of the cutter and thepressure angles can then be expressed in the form of parametric vector equations. For various types ofconjugate cams, the equations for such expressions are formulated, and examples are provided toillustrate the approach. The procedure is especially simple to program.Keywords:conjugate cam pro? le, instant centre, vectorNOTATIONAcontact pointBcontact pointCroller centreddistance between the roller centres,breadth of the ? at-faced followerDroller centreefollower offsetEpointfdistance from the cam centre to thefollower pivot pointGcutter centreHcutter centreI12, I13,I23instant centresiunit vectorjunit vectorlarm length of the followerLdistance of the follower centre from thecam centre measured parallel to the rollertranslation LyO2? xed pivot of the camO3? xed pivot of the oscillating followerqdistance from the cam centre to theinstant centre I23Qlocation of the instant centre I23rbbase circle radiusrccutter radiusrfradius of the roller followerSfollower motion program SyttimeVQspeed of point Q(X,Y)Cartesian coordinate system ? xed on thecamaA,aBanglesZsubtending angle of the follower armsycam rotation anglexAangular displacement function of thefollower xAyfA,fBpressure angleso2angular velocity of the cam1INTRODUCTIONIn a cam mechanism, the follower must always be heldin contact with the cam throughout the motion cycle,and this is usually accomplished by a positive drive or areturn spring. A normal conjugate cam mechanism caneliminate the return-spring force and thus result in lowercontact stresses, when compared with the spring-loadedtype. This important advantage makes it especiallysuitable for high-speed applications. To perform safelyand reliably its intended function, however, the con-jugate cams must be properly designed and accuratelymanufactured. Therefore, the cam pro? les and the pathsof the cutter centre should be determined analytically.Hanson and Churchill 1, employing the theory ofenvelopes, presented an analytical method for comput-ing disccam pro? le coordinates. Although the theory ofenvelopes is not always taught in the college course ofcalculus, this method has been widely adopted. On theother hand, Davidson 2 suggested another methodThe MS was received on 9 September 2002 and was accepted afterrevision for publication on 4 August 2003.1117C11702# IMechE 2003Proc. Instn Mech. Engrs Vol. 217 Part C:J. Mechanical Engineering Science at Shanghai Jiaotong University on January 13, 2013pic.sagepub.comDownloaded from using instant centres, but his contribution seems tohave attracted little attention from kinematicians. As amatter of fact, the analytical method using the instantcentre approach can provide a convenient means fordetermining the disccam pro? le and cutter coordi-nates. In addition, it is applicable to not only thecommon spring-loaded type but also the conjugatecam mechanism.2CONJUGATE CAMS WITH AN OFFSETTRANSLATING ROLLER FOLLOWERFigure 1 shows a conjugate cam mechanism with anoffset translating roller follower. There are two cams Aand B, ? xed on a common shaft. Two follower rollers Cand D, mounted to a common follower, are each pushedin opposite directions by the conjugate cams. Setting upa Cartesian coordinate system (X,Y) ? xed on the camand with its origin at the ? xed pivot O2, the cam pro? lecoordinates may be expressed in terms of the camrotation angle y, which is measured against the directionof cam rotation from the reference radial to cam centre-line parallel to roller translation.A conjugate cam mechanism may be considered as apermanent critical form and must always have threevelocity instant centres 3. As shown in Fig. 1, thismeans that the two normal lines through the points ofcontact and the line of centres must always intersect at acommon point, the instant centre I23, where I denotesthe instant centre and subscripts indicate the relatedlinks. For simplicity, in the following, the ground linkwill be consistently numbered as 1, the cam as 2 andthe follower as 3. For the sake of clarity, two otherinstant centres I12and I13are also located and labelledin the ? gure. By labelling instant centre I23as Q andO2Q q, the speed of point Q on the cam can beexpressed asVQ qo21where o2is the angular velocity of the cam. In order tolet y have a counterclockwise angle, in this paper, thecam is to rotate clockwise.On the other hand, for a translating follower, allpoints on the follower have the same velocity. Therefore,the speed of point Q on the follower can be expressed asVQdLydtdLydydydtdLydyo22where Ly is the displacement function of the follower:Ly ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?rb rf2 e2q Sy3where rbis the base circle radius of cam A, rfis theradius of the roller follower, e is the offset and Sy isthe follower motion program. (Since the cam is to rotateclockwise, the quantity e is negative if the offset is to theright; in the position shown it is positive.) By de? nitionof the instant centre, instant centre I23(point Q) is apoint commontolinks 2 (conjugate cam) and3(follower) having the same velocity. Therefore, fromequations (1) and (2),q dLydydSydy4As a result, after rb,rf, e and Sy have been selected, foreach speci?ed value of y, the roller centre C may belocated by application of equation (3) and point Q byapplication of equation (4).The pressure angle is the angle between the commonnormal at the contact point and the direction of motionof the follower 4. For cam A, it is the angle betweenlines CQ and CE. From 4CQE, the pressure angle fAof cam A can be expressed asfA tan1q eLy tan11LydSydy e?5Therefore, the parametric equations for the pro? leFig. 1Conjugate disc cams with an offset translating rollerfollowerL-I WU1118Proc. Instn Mech. Engrs Vol. 217 Part C:J. Mechanical Engineering ScienceC11702# IMechE 2003 at Shanghai Jiaotong University on January 13, 2013pic.sagepub.comDownloaded from coordinates of cam A areO2A O2E EC CA6whereO2E ecosy 90i esiny 90j7EC Lycosyi Lysin yj8CA rfcosy 180 fAi rfsiny 180 fAj9In the same manner, after the distance between rollercentres d has been selected, the other roller centre D maybe located. From 4DQE, the pressure angle fBof camB can be expressed asfB tan1q ed Ly tan11d LydSydy e?10Similarly, the parametric equations for the pro? lecoordinates of cam B areO2B O2E ED DB11whereED d Lycosy 180i d Lysiny 180j12DB rfcosy fBi rfsiny fBj13In practice, the cutter or grinding wheel is frequentlychosen larger than the follower roller for reasonablegrinding wear life 5. The locations of the cutter centresfor cutting cams A and B are also shown in Fig. 1.Because the cutter and roller centres must lie on acommonnormal tothe campro? le 5, normallyoutward extending the cam pro? le by a length of cutterradius rcobtains the location of the cutter centre. Inother words, for cam A, the cutter centre G and pointsQ, A and C must always lie on a line. Therefore, theparametric equations for the coordinates of the cuttercentre G areO2G O2E EC CG14whereCG rc rfcosy fAi rc rfsiny fAj15The location of the cutter centre H for cutting cam B canalso be located in the same way:O2H O2E ED DH16whereDH rc rfcosy 180 fBi rc rfsiny 180 fBj17In fact, the conjugate cam pro? les of Fig. 1 have beendrawn, by applying these equations,tomeetthefollowing requirements. The follower is to rise 20mmwith cycloidal motion while the cam rotates clockwisefrom 08 to 1008, dwell for the next 508, return withcycloidal motion for 1008 cam rotation and dwell for theremaining 1108. Both follower rollers have the sameradius of 10mm. The offset e is 12mm and the distancebetween the roller centres, d, is 113mm. The base circleradius of cam A is 40mm.3CONJUGATE CAMS WITH A TRANSLATINGFLAT-FACED FOLLOWERFigure 2 shows a conjugate cam mechanism with atranslating ? at-faced follower. Setting up a Cartesiancoordinate system (X, Y ) ?xed on the cam and with itsFig. 2Conjugate disc cams with a translating ? at-facedfollowerCALCULATING CONJUGATE CAM PROFILES BY VECTOR EQUATIONS1119C11702# IMechE 2003Proc. Instn Mech. Engrs Vol. 217 Part C:J. Mechanical Engineering Science at Shanghai Jiaotong University on January 13, 2013pic.sagepub.comDownloaded from origin at the ? xed pivot O2, the cam pro? le coordinatesmay be expressed in terms of y.Thismechanismmayalsobeconsideredasapermanent critical form and must always have threeinstant centres. It follows that the two normal lines andthe line of centres must always intersect at a commonpoint, the instant centre I23. In this case, it also meansthat the two contact points A and B and instant centreI23must always lie on a vertical line. By labelling theinstant centre I23as Q and O2Q q, the speed of pointQ on the cam can be expressed asVQ qo218The speed of point Q on the follower can be expressedasVQdLydtdLydydydtdLydyo219where Ly is the displacement function of the follower:Ly rb Sy20where rbis the base circle radius of cam A and Sy isthe follower motion program. From equations (18) and(19),q dLydydSydy21As a result, after rband Sy have been selected, for eachspeci?ed value of y, point Q may be located by means ofthe corresponding value of q, which can be found fromequation (21), and then the contact point A may belocated by means of the corresponding value of Ly.Therefore, the parametric equations for the pro? lecoordinates of cam A areO2A O2Q QA22whereO2Q qcosy 90i qsiny 90 j23QA Lycosyi Lysin yj24Similarly, the parametric equations for the pro? lecoordinates of cam B areO2B O2Q QB25whereQB d Lycosy 180i d Lysiny 180j26and d is the breadth of the follower.As previously indicated, normally outward extendingthe cam pro? le by a length of cutter radius rcobtains thelocation of the cutter centre. Therefore, for cam A, theparametric gt ectiothe,tgwhich is measured against the direction of cam rotationfrom the reference radial on the cam to the line betweenthe cam centre and follower pivot point.Because it is a permanent critical form, its two normallines and the line of centres must always intersect at theinstant centre I23, point Q. The speed of point Q on thecam can be expressed asVQ qo231where q O2Q and o2is the angular velocity of thecam. On the other hand, the speed of point Q on thefollower can be expressed asVQ f qdxAydt f qdxAydyo232where xAy is the angular displacement function of thefollower A:xAy cos1l2 f2 rb rf22lf# Sy33where rbis the base circle radius of cam A, rfis theradius of the roller follower and Sy is the followerangular motion program. From equations (31) and (32)and after some algebraic manipulation,q fdxAydy1 dxAydyfdSydy1 dSydy34From 4O3QC and the cosine law,QC ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?l2 f q2 2lf qcosxAyq35From 4O3QD and the cosine law,QD ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?l2 f q2 2lf qcos Z xAyq36where Z is the subtending angle of the follower arms.From 4O3QC and the sine law,aA sin1l sin xAyQC37From 4O3QD and the sine law,aB sin1l sinZ xAyQD?38Therefore, the parametric equations for the pro? lecoordinates of cam A areO2A O2Q QA39whereO2Q qcosy 180i qsiny 180j40QA QC rfcosy aAi QC rfsiny aAj41The parametric equations for the pro? le coordinates ofcam B areO2B O2Q QB42whereQB QD rfcosy aBi QD rfsiny aBj43From 4O3QC, the pressure angle fAof cam A can beexpressed asfA 90 aA xAy44From 4O3QD, the pressure angle fBof cam B can beexpressed asfB 90 aB Z xAy45As previously indicated, the cutter and roller centremust lie on a common normal to the cam pro? le.Therefore, normally outward extending the cam pro? leby a length of cutter radius rcobtains the location of thecutter centre. In other words, for cam A, the cuttercentre G and points Q, A and C must always lie on aline. As a result, the parametric equations for thecoordinates of the cutter centre G areO2G O2Q QG46whereQG QC rf rccosy aAi QC rf rcsiny aAj47The location of the cutter centre H for cutting cam B canalso be located in the same way:O2H O2Q QH48whereQH QD rf rccosy aBi QD rf rcsiny aBj49The conjugate cam pro?les of Fig. 3 have been drawn,by applying these equations, to meet the followingrequirements. The follower is to oscillate 308 clockwisewith cycloidal motion while the cam rotates clockwisefrom 08 to 1208, dwell for the next 408, return withcycloidal motion for 1208 cam rotation and dwell for theremaining 808. The distance between pivots, f, is 80mm.CALCULATING CONJUGATE CAM PROFILES BY VECTOR EQUATIONS1121C11702# IMechE 2003Proc. Instn Mech. Engrs Vol. 217 Part C:J. Mechanical Engineering Science at Shanghai Jiaotong University on January 13, 2013pic.sagepub.comDownloaded from Both follower arms have the same length of 32mm andboth follower rollers have the same radius of 16mm.The base circle radius of cam A, rb, is 57.32mm and thesubtending angle of the follower arms, Z, is 1008.5CONJUGATE CAMS WITH AN OSCILLATINGFLAT-FACED FOLLOWERFigure 4 shows a conjugate cam mechanism with anoscillating ? at-faced follower. In this case, f representsthe distance from the cam centre to the follower pivotpoint and e represents the follower face offset from thefollower pivot point. (The quantity e is positive in Fig. 4.If the follower face is offset from the pivot point towardsthe cam centre, it is negative.) Setting up a Cartesiancoordinate system (X,Y) ?xed on the cam and with itsorigin at the ? xed pivot O2, the cam pro? le coordinatesmay be expressed in terms of y.This is also a permanent critical form; its two normallines and the line of centres must always intersect at theinstant centre Q. The speed of point Q on the cam can beexpressed asVQ qo250where q O2Q. The speed of point Q on the followercan be expressed asVQ f qdxAydt f qdxAydyo251where xAy is the angular displacement function of thefollower A:xAy sin1rb ef? Sy52where rbis the base circle radius of cam A, e is the offsetof the follower and Sy is the follower angular motionprogram. From equations (50) and (51) and after somealgebraic manipulation,q fdxAydy1 dxAydyfdSydy1 dSydy53From 4O3QC,QC f q sin xAy54aA 90 xAy55From 4O3QD,QD f qsin Z xAy56aB 90 Z xAy57where Z is the subtending angle of the follower arms.Therefore, the parametric equations for the pro? lecoordinates of cam A areO2A O2Q QA58whereO2Q qcosy 180i qsiny 180j59QA QC ecosy aAi QC esiny aAj60The parametric equations for the pro? le coordinates ofcam B areO2B O2Q QB61whereQB QD ecosy aBi QD esiny aBj62From 4O3AC, the pressure angle fAof cam A can beFig. 4Conjugate disc cams with an oscillating ? at-facedfollowerL-I WU1122Proc. Instn Mech. Engrs Vol. 217 Part C:J. Mechanical Engineering ScienceC11702# IMechE 2003 at Shanghai Jiaotong University on January 13, 2013pic.sagepub.comDownloaded from expressed asfA tan1ef qcosxAy63From 4O3BD, the pressure angle fBof cam B can beexpressed asfB tan1ef qcosZ xAy?64As previously indicated, normally outward extendingthe cam pro? le by a length of cutter radius rcobtains thelocation of the cutter centre. Therefore, the parametricequations for the coordinates of the cutter centre G forcutting cam A areO2G O2Q QG65whereQG QC e rccosy aAi QC e rcsiny aAj66The location of the cutter centre H for cutting cam B canalso be located in the same way:O2H O2Q QH67whereQH QD e rccosy aBi QD e rcsiny aBj68The conjugate cam pro? les of Fig. 4 have been drawn,by applying these equations, to meet the followingrequirements. The follower is to oscillate 158 clockwisewith cycloidal motion while the cam rotates clockwisefrom 08 to 1208, dwell for the next 508, return withcycloidal motion for 1008 cam rotation and dwell for theremaining 908. The distance between pivots, f, is 80mm.Both follower arms have the same offset of 14mm. Thebase circle radius of cam A, rb, is 40mm and thesubtending angle of the follower arms, Z, is 508.6DISCUSSION AND CONCLUSIONAnother topic that is frequently encountered in the camdesign is the determination of cam curvature. Thedesigned cam pro? le may have distortion under certainconditions. However, after the parametric equationsdescribing the cam pro? le have been developed, thecurvature of the cam pro? le can be calculated accurately57. When undercutting occurs, the radius of curvatureswitches sign from positive to negative. As a conse-quence, the potential distortion of the cam pro? le maybe checked analytically.A conjugate cam mechanism may be considered as apermanent critical form, and the two normal linesthrough the points of contact and the line of centresmust always intersect at the instant centre I23. For a cammechanism with speci?ed system parameters, followerdimensions and camfollower motion program, thethree instant centres of the mechanism can be located.According to the locations of instant centres andfollower position, the contact points between the camand the follower, the pressure angles and the locationsof the cutter centre can be determined and expressed inthe form of parametric vector equations.ACKNOWLEDGEMENTThe author is grateful to the National Science Councilof the Republic of China for supporting this researchunder Grant NSC 92-2212-E-007-056.REFERENCES1 Hanson, R. S. and Churchill, F. T. Theory of envelopesprovides new cam design equations. Product Engng, 20August 1962, 4555.2 Davidson, J. K. Calculating cam pro? les quickly. Mach.Des., 7 December 1978, 151155.3 Paul, B. Kinematics and Dynamics of Planar Machinery,1979, pp. 285289 (Prentice-Hall, Englewood Cliffs, NewJersey).4 Martin, G. H. Kinematics and Dynamics of Machines, 2ndedition, 1982, p. 212 (McGraw-Hill, New York).5 Rothbart, H. A. Cams: Design, Dynamics, and Accuracy,1956, pp. 7889 (John Wiley, New York).6 Molian, S. The Design of Cam Mechanisms and Linkages,1968, pp. 6470 (American Elsevier, New York).7 Jensen, P. W. Cam Design and Manufacture, 2nd edition,1987, pp. 94105 (Marcel Dekker, New York).CALCULATING CONJUGATE CAM PROFILES BY VECTOR EQUATIONS1123C11702# IMechE 2003Proc. Instn Mech. Engrs Vol. 217 Part C:J. Mechanical Engineering Science at Shanghai Jiaotong University on January 13, 2013pic.sagepub.comDownloaded from