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An analytical design for three circular-arc cams Chiara Lanni, Marco Ceccarelli * , Giorgio Figliolini Dipartimento di Meccanica, Strutture, Ambiente e Territorio, UniversitC18a di Cassino, Via Di Biasio 43, 03043 Cassino (Fr), Italy Received 10 July 2000; accepted 22 January 2002 Abstract In this paper we have presented an analytical description for three circular-arc cam proles. An ana- lytical formulation for cam proles has been proposed and discussed as a function of size parameters for design purposes. Numerical examples have been reported to prove the soundness of the analytical design procedure and show the engineering feasibility of suitable three circular-arc cams. C211 2002 Elsevier Science Ltd. All rights reserved. 1.Introduction A cam is a mechanical element, which is used to transmit a desired motion to another me- chanical element by direct surface contact. Generally, a cam is a mechanism, which is composed of three dierent fundamental parts from a kinematic viewpoint 1,2: a cam, which is a driving element; a follower, which is a driven el- ement and a xed frame. Cam mechanisms are usually implemented in most modern applications and in particular in automatic machines and instruments, internal combustion engines and control systems 3. Cam and follower mechanisms can be very cheap, and simple. They have few moving parts and can be built with very small size. The design of cam prole has been based on simply geometric curves, 4, such as: parabolic, harmonic, cycloidal and trapezoidal curves 2,5 and their combinations 1,2,6,7. In this paper we have addressed attention to cam proles, which are designed as a collection of circular arcs. Therefore they are called circular-arc cams 5,8. * Corresponding author. E-mail address: ceccarelliing.unicas.it (M. Ceccarelli). 0094-114X/02/$ - see front matter C211 2002 Elsevier Science Ltd. All rights reserved. PII: S0094-114X(02)00032-0 Mechanism and Machine Theory 37 (2002) 915924 Circular-arc cams can be easily machined and can be used in low-speed applications 9. In addition, circular-arc cams could be used for micro-mechanisms and nano-mechanisms since very small manufacturing can be properly obtained by using elementary geometry. An undesirable characteristic of this type of cam is the sudden change in the acceleration at the prole points where arcs of dierent radii are joined 5. A limited number of circular-arcs is usually advisable so that the design, construction and operation of cam transmission can be not very complicated and they can become a compromise for simplicity and economic characteristics that are the basic advantages of circular-arc cams 8. Recently new attention has been addressed to circular-arc cams by using descriptive viewpoint 10, and for design purposes 11,12. In this paper we have described three circular-arc cams by taking into consideration the geo- metrical design parameters. An analytical formulation has been proposed for three circular-arc cams as an extension of a formulation for two circular-arc cams that has been presented in a previous paper 12. 2.Ananalyticalmodelforthreecircular-arccams Ananalyticalformulationcanbeproposedforthreecircular-arccamsinagreementwithdesign parameters of the model shown in Figs. 1 and 2. Signicantparametersforamechanicaldesignofathreecircular-arccamare:Fig.18;therise angle a s , the dwell angle a r , the return angle a d , the action angle a a a s a r a d , the maximum lift h 1 . Fig. 1. Design parameters for general three circular-arc cams. 916 C. Lanni et al. / Mechanism and Machine Theory 37 (2002) 915924 The characteristic loci of a three circular-arc cams are shown in Fig. 2as: the rst circle C 1 of the cam prole with q 1 radius and centre C 1 ; the second circle C 2 of the cam prole with q 2 radius and centre C 2 ; the third circle C 3 of the cam prole with q 3 radius and centre C 3 ; the base circle C 4 withradius r andthe centreis O; the liftcircle C 5 ofthe camprolewith (r h 1 ) radiusandcentre O; the roller circle with radius q centred on the follower axis. In addition signicant points are: D C17x D ;y D which is the point joining C 1 with C 5 ; F C17x F ;y F which is the point joining C 1 with C 3 ; G C17x G ;y G which is the point joining C 3 with C 2 ; A C17x A ;y A ) which is the point joining C 2 with C 4 . x and y are Cartesian co-ordinates of points with respect to the xed frame OXY, whose origin O is a point of the cam rotation axis. Additional signicant loci are: t 13 which is the co- incidenttangentialvectorbetween C 1 and C 3 ; t 15 whichisthecoincidenttangentialvectorbetween C 1 and C 5 ; t 23 which is the coincident tangential vector between C 2 and C 3 ; t 24 which is the co- incident tangential vector between C 2 and C 4 . The model shown in Figs. 1 and 2can be used to deduce a formulation, which can be useful both for characterizing and designing three circular-arc cams. Analytical description can be proposed when the circles are formulated in the suitable form: circle C 1 with radius q 2 1 x 1 C0 x F 2 y 1 C0y F 2 passing through point F as x 2 y 2 C02xx 1 C02yy 1 C0 x 2 F C0y 2 F 2x 1 x F 2y 1 y F 0 1 circle C 2 with radius q 2 2 x 2 C0 x A 2 y 2 C0y A 2 passing through point A as x 2 y 2 C02xx 2 C02yy 2 C0 x 2 A C0y 2 A 2x 2 x A 2y 2 y A 0 2 circle C 2 with radius q 2 2 x 2 C0 x G 2 y 2 C0y G 2 passing through point G as x 2 y 2 C02xx 2 C02yy 2 C0 x 2 G C0y 2 G 2x 2 x G 2y 2 y G 0 3 Fig. 2. Characteristic loci for three circular-arc cams. C. Lanni et al. / Mechanism and Machine Theory 37 (2002) 915924 917 circle C 3 with radius q 2 3 x 3 C0 x F 2 y 3 C0y F 2 passing through point F as x 2 y 2 C02xx 3 C02yy 3 C0 x 2 F C0y 2 F 2x 3 x F 2y 3 y F 0 4 circle C 3 with radius q 2 3 x 3 C0 x G 2 y 3 C0y G 2 passing through point G as x 2 y 2 C02xx 3 C02yy 3 C0 x 2 G C0y 2 G 2x 3 x G 2y 3 y G 0 5 circle C 4 with radius r as x 2 y 2 r 2 6 circle C 5 with radius (r h 1 )as x 2 y 2 r h 1 2 7 Additional characteristic conditions can be expressed in the form as therstcircleC 1 andliftcircleC 5 musthavethesametangentialvector t 15 atpointDexpressedas xx 1 yy 1 C0 x 1 x D C0y 1 y D 0 8 the base circle C 4 and second circle C 2 must have the same tangential vector t 24 at point A ex- pressed as xx 2 yy 2 C0 x 2 x A C0y 2 y A 0 9 the second circle C 2 and third circle C 3 must have the same tangential vector t 23 at point G ex- pressed as xx 3 C0 x 2 yy 3 C0y 2 x 3 x G y 3 y G C0 x 1 x G C0y 1 y G 0 10 the rst circle C 1 and the second circle C 2 must have the same tangential vector t 12 at point F expressed as xx 1 C0 x 3 yy 1 C0y 3 x 3 x F y 3 y F C0 x 1 x F C0y 1 y F 0 11 Eqs. (1)(11) may describe a general model for three circular-arc cams and can be used to draw the mechanical design as shown in Fig. 2. 3.Ananalyticaldesignprocedure Eqs. (1)(11) can be used to deduce a suitable system of equations, which allows solving the co- ordinates of the points C 1 , C 2 , C 3 , F and G when suitable data are assumed. It is possible to distinguish four dierent design cases by using the proposed analytical de- scription. In a rst case we can consider that the numeric value of the parameters h 1 , r, a s , a r , a d , q 1 , q 2 , and co-ordinates of the points A, C 1 , C 2 , D and G are given, and the co-ordinates of points C 3 , F aretheunknowns.Whentheactionangle a a isequalto180C176,theco-ordinatex A ofpoint A isequal to zero. Since A is the point joining C 2 and C 4 then the centre C 2 of the second circle C 2 lies on the Y axisandthereforetheco-ordinate x 2 ofthecentre C 2 is equaltozero.ByusingEqs.(1)(11)itis 918 C. Lanni et al. / Mechanism and Machine Theory 37 (2002) 915924 possible to deduce a suitable system of equations which allows to solve the co-ordinates of the points C 3 and F. Analytical formulation can be expressed by means of the following conditions: the rst circle C 1 passing across points F and D in the form x F C0 x 1 2 y F C0y 1 2 x D C0 x 1 2 y D C0y 1 2 12 the third circle C 3 passing across points F and G in the form x F C0 x 3 2 y F C0y 3 2 x G C0 x 3 2 y G C0y 3 2 13 coincident tangents to C 1 and C 3 at the point F in the form x 3 C0 x 1 y 3 C0y 1 x F C0 x 3 y F C0y 3 14 coincident tangents to C 2 and C 3 at the point G in the form x 2 C0 x 3 y 2 C0y 3 x G C0 x 2 y G C0y 2 15 When x 2 x A 0 are assumed, Eqs. (12)(15) can be expressed as x 2 F y 2 F C02x 1 x F C02y 1 y F C0 x 2 D C0y 2 D 2x 1 x D 2y 1 y D 0 x 2 F y 2 F C02x 3 x F C02y 3 y F C0 x 2 G C0y 2 G 2x 3 x G 2y 3 y G 0 x F C0 x 3 y 3 C0y 1 C0 x 3 C0 x 1 y F C0y 3 0 x G y 2 C0y 3 C0 x 3 y G C0y 2 0 16 If the position of the centre C 2 is unknown and the direction of the centre C 1 lies on the OD straight line, we can approach referring to Fig. 2a second problem: namely the value of the parameters h 1 , r, a s , a r , a d , q 1 , and the co-ordinates of the points C 2 , A, D and G are known and the co-ordinates of the points C 1 , F and C 3 are unknown. Again we may assume a a 180C176 and consequently x A x 2 0. Two additional conditions are necessary to have a solvable system together with Eq. (9). They are the second circle C 2 passing across points G and A in the form x G C0 x 2 2 y G C0y 2 2 x A C0 x 2 2 y A C0y 2 2 17 straight-line containing points O, A and C 2 in the form x 2 y A C0 x A y 2 0 18 Thus, the second case can be solved by Eqs. (16)(18). If the position of the centre C 1 is unknown but we know that it lies on the OD straight line, we can approach a third design problem: namely the value of the parameters h 1 , r, a s , a r , a d , q 1 , and the co-ordinates of the points A, D and G are known and the co-ordinates of the points C 1 , C 2 , F and C 3 are unknown. Again we may assume a a 180C176 and consequently x A x 2 0. Two ad- ditional conditions are necessary to have a solvable system together with Eqs. (16)(18). They are C. Lanni et al. / Mechanism and Machine Theory 37 (2002) 915924 919 the rst circle C 1 passing across point D in the form x D C0 x 1 2 y D C0y 1 2 q 2 1 19 straight-line containing points O, D and C 1 in the form x D y 1 C0 x 1 y D 0 20 Finally we may approach the fourth case when a a 180C176 and x A 6 0 and also x 2 6 0. Referring to Fig. 1, in which a a is the angle between the general position of the point A and the Y axis, the value of the parameters h 1 , r, a s , a r , a d , q 1 , and the co-ordinates of the points A, D and G are known and the co-ordinates of the points C 1 , C 2 , C 3 and F are unknown. The fourth of Eq. (16) can be expressed as x 2 C0 x 3 y G C0y 2 C0y 2 C0y 3 x G C0 x 2 0 21 Thus, the general design case can be solved by using Eqs. (12)(14) and Eqs. (17)(21). A design procedure can be obtained by using the above-mentioned formulation in order to compute the design parameters. In particular, the proposed formulation has been useful for a design procedure which makes use of MAPLE to solve for the design unknowns. 4.Numericalexamples Severalnumericexampleshavebeensuccessfullycomputedinordertoprovethesoundnessand numerical eciency of the proposed design formulation. It has been found that only one solution can represent a signicant circular-arc cam design for any of the formulated design cases. In the Example 1 of Fig. 3 referring to the rst design case, the data are given as h 1 15 mm, r 40 mm, a r 40C176, a s a d 70C176, q 1 17 mm, A C170;40 mm, D C1751:68 mm; 18:81 mm C 1 C1735:71 mm; 13:00 mm, C 2 C170mm; C075:64 mm and G C1722:24mm; 37:84 mm. Fig. 3 shows results for the design case, which has been formulated by Eq. (16). In particular, Fig. 3(a) shows the rst solution of the analytical formulation. We can note that points F, C 1 and C 3 are aligned inthe order F, C 1 and C 3 and points G, C 3 and C 2 in the order G, C 3 and C 2 respectively to the rst and second arcs cam prole. Fig. 3(b) shows the second solution of the analytical for- mulation.Acamprolecannotbeidentiedsince F pointdoesnotliealsooncircle C 1 .Signicant points F, C 1 and C 3 are aligned in the same order with respect to the case in Fig. 3(a); points G, C 2 and C 3 arealignedinthe C 2 , G and C 3 sequentialorderwhichisdierentrespecttothecaseinFig. 3(a)anddonotgiveacamprole.Fig.3(c)showsthethirdsolutionofanalyticalformulationthat issimilartothecaseofFig.3(b).Fig.3(d)showsthefourthsolutionofanalyticalformulation.We can note that in correspondence of point D there is a cusp. In addition, points F and G are very near to centre C 3 so that a sudden change of curvature is obtained in the cam prole as shown in Fig.3(d).ThusapracticalfeasibledesignisrepresentedonlybyFig.3(a)thatcanbecharacterised by the proper order F, C 1 and C 3 and G, C 3 and C 2 of the meaningful points. The feasible numerical solution in Fig. 3(a) is characterised by the values: x F 46:78 mm, y F 25:91 mm, x 3 11:99 mm, y 3 C014:47 mm. 920 C. Lanni et al. / Mechanism and Machine Theory 37 (2002) 915924 In the Example 2of Fig. 3 the data are given as h 1 15 mm, r 40 mm, a r 40C176, a s a d 70C176, q 1 17 mm, A C170; 40 mm, D C1751:68 mm; 18:81 mm, C 1 C1735:71 mm; 13:00 mm and G C1722:24mm; 37:84 mm. In this case Fig. 3 represents also the design solution which has been obtained by using Eqs. (16)(18) for the second design case. The feasible numerical solution is characterised by the values: x F 46:78 mm, y F 25:91 mm, x 3 11:99 mm, y 3 C014:47 mm, x 2 0 mm, y 2 C075:64 mm. In the Example 3 of Fig. 4 referring to the third design case the data are given as h 1 15 mm, r 40 mm, a r 40C176, a s a d 70C176, q 1 17 mm, A C170; 40 mm, D C1751:68 mm; 18:81 mm and G C1722:24 mm; 37.84 mm). Fig. 4 shows results for the design case, which has been formulated by Eqs. (16)(20). Fig. 4(a) shows the rst solution of analytical formulation. This case is similar to the solution represented in Fig. 3(d). Fig. 4(b) shows the second solutionof analytical formulation. We can note that point F is located below point D so that points F, C 1 and C 3 are not aligned. Fig. 3(c) shows the third Fig. 3. Examples 1 and 2: graphical representation of design solutions for Eq. (16) and design solutions for Eqs. (16) (18). Only case (a) is a practical feasible design. C. Lanni et al. / Mechanism and Machine Theory 37 (2002) 915924 921 solution of analytical formulation, which is the same of the case reported in Fig. 3(a). Thus a practical feasible design is represented only by Fig. 4(c). The feasible numerical solution is characterised by the values: x F 46:78 mm, y F 25:91 mm, x 3 11:99 mm, y 3 C014:47 mm, x 2 0 mm, y 2 C075:64 mm, x 1 35:71 mm, y 1 13:00 mm. IntheExample4ofFig.5referringtothefourthdesigncase,thedataare givenas h 1 15mm, r 40 mm, a r 40C176, a s a d 70C176, q 1 17 mm, A C173:48 mm; 39.84 mm), D C1751:68 mm; 18.81 mm) and G C1722:24 mm; 37.84 mm). Fig. 5 shows results for the design case, which has been formulated by Eqs. (16)(21). Fig. 5(a) shows the rst solution of the analytical formulation. This design is similar to the case reported in Fig. 4(a), but the location of point C 1 is dierent. Points F, C 1 and C 3 are aligned in the C 3 , F and C 1 order. Fig. 5(b) shows the second solution of analytical formulation, which is similar to the case in Fig. 4(a). Fig. 5(c) shows the third solution of analytical formulation. This case shows a Fig. 4. Example 3: graphical representation of design solutions for Eqs. (16)(20). Only case (c) is a practical feasible design. 922 C. Lanni et al. / Mechanism and Machine Theory 37 (2002) 915924 solution, which is similar to the case reported in Fig. 4(c). Thus a practical feasible design is represented only by Fig. 5(c). The feasible numerical solution is characterised by the values: x F 48:15 mm, y F 24:58 mm, x 3 16:92mm, y 3 C04:50 mm, x 2 C040:01 mm, y 2 C0457:26 mm, x 1 35:71 mm, y 1 13:00 mm. 5.Applications A novel interest can be addressed to approximate design of cam proles for both new design purposes and manufacturing needs. Analytical design formulation is required to obtain ecient design algorithms. In addition, closed-form formulation can be also useful to characterise cam proles in both analysis proce- dures and synthesis criteria. The approximated proles with circular-arcs can be of particular interest also to obtain analytical expressions for kinematic characteristics of any proles that can be approximated by segments of proper circular arcs. Fig. 5. Example 4: graphical representation of design solutions for Eqs. (16)(21). Only case (c) is a practical feasible design. C. Lanni et al. / Mechanism and Machine Theory 37 (2002) 915924 923 Indeed, the circular-arc cam proles have become of current interest because of applications in mini-mechanisms and micro-mechanisms. In fact, when the size of a mechanical design is reduced to the scale of millimeters (mini-mechanisms) and even micron (micro-mechanisms) the manu- facturing of polynomial cam prole becomes dicult and even more complicated is a way to verify it. Therefore, it can be convenient to design circular-arc cam proles that can be also easily tested experimentally. In addition, stronger and stronger demand of low-cost automation is giving new interest to approximatedesigns,whichcanbeusedonly forspecic tasks.Thisis the caseof circular-arccam proles that can be conveniently used in low speed machinery or in low-precision applications. 6.Conclusions In this paper we have proposed an analytical formulation which describes the basic design characteristics of three circular-arc cams. A design algorithm has been deduced from the for- mulation, which solves design problems with great numerical eciency. Numerical examples have been reported in the paper to show and discuss the multiple design solutions and the engineering feasibility of three circular-arc cams. References 1 F.Y. Chen, Mechanics and Design of Cam Mechanisms, Pergamon Press, New York, 1982. 2 J. Angeles, C.S. Lopez-Cajun, Optimization of Cam Mechanisms, Kluwer Academic Publishers, Dordrecht, p. 1991. 3 R. Norton, Cam and cams follower (Chapter 7), in: G.A. Erdman (Ed.), Modern Kinematics: Developments in the Last Forty Years, Wiley-Interscience, New York, 1993. 4 F.Y. Chen, A survey of the state of the art of cam system dynamics, Mechanism and Machine Theory 12(1977) 201224. 5 G. Scotto Lavina, in: Sistema (Ed.), Applicazioni di Meccanica Applicata alle Macchine, Roma, 1971. 6 H.A. Rothbar, Cams Design, Dynamics and Accuracy, Wiley, New York, 1956. 7 J.E. Shigley, J.J. Uicker, Theory of Machine and Mechanisms, McGraw-Hill, New York, 1981. 8 P.L. Magnani, G. Ruggieri, Meccanismi per Macchine Automatiche, UTET, Torino, 1986. 9 N.P. Chironis, Mechanisms and Mechanical Devices Sourcebook, McGraw-Hill, New York, 1991. 10 V.F. Krasnikov, Dynamics of cam mechanisms with cams countered by segments of circles, in: Proceedings of the International Conference on Mechanical Transmissions and Mechanisms, Tainjin, 1997, pp. 237238. 11 J.Oderfeld,A.Pogorzelski,Ondesigningplanecammechanisms, in:ProceedingsoftheEighthWorldCongresson the Theory of Machines and Mechanisms, Prague, vol. 3, 1991, pp. 703705. 12 C. Lanni, M. Ceccarelli, J.C.M. Carvhalo, An analytical design for two circular-arc cams, in: Proceedings of the Fourth Iberoamerican Congress on Mechanical Engineering, Santiago de Chile, vol. 2, 1999. 924 C. Lanni et al. / Mechanism and Machine Theory 37 (2002) 915924