喜歡就充值下載吧。。資源目錄里展示的文件全都有，，請放心下載，，有疑問咨詢QQ:414951605或者1304139763 ======================== 喜歡就充值下載吧。。資源目錄里展示的文件全都有，，請放心下載，，有疑問咨詢QQ:414951605或者1304139763 ======================== An analytical design for three circular-arc camsChiara Lanni, Marco Ceccarelli*, Giorgio FiglioliniDipartimento di Meccanica, Strutture, Ambiente e Territorio, Universit? a a di Cassino, Via Di Biasio 43,03043 Cassino (Fr), ItalyReceived 10 July 2000; accepted 22 January 2002AbstractIn this paper we have presented an analytical description for three circular-arc cam profiles. An ana-lytical formulation for cam profiles has been proposed and discussed as a function of size parameters fordesign purposes. Numerical examples have been reported to prove the soundness of the analytical designprocedure and show the engineering feasibility of suitable three circular-arc cams.? 2002 Elsevier Science Ltd. All rights reserved.1. IntroductionA 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 different fundamental parts froma kinematic viewpoint 1,2: a cam, which is a driving element; a follower, which is a driven el-ement and a fixed frame. Cam mechanisms are usually implemented in most modern applicationsand in particular in automatic machines and instruments, internal combustion engines andcontrol systems 3.Cam and follower mechanisms can be very cheap, and simple. They have few moving parts andcan be built with very small size.The design of cam profile 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 profiles, which are designed as a collection ofcircular 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 ? 2002 Elsevier Science Ltd. All rights reserved.PII: S0094-114X(02)00032-0Mechanism and Machine Theory 37 (2002) cams can be easily machined and can be used in low-speed applications 9. Inaddition, circular-arc cams could be used for micro-mechanisms and nano-mechanisms since verysmall 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 theprofile points where arcs of different radii are joined 5.A limited number of circular-arcs is usually advisable so that the design, construction andoperation of cam transmission can be not very complicated and they can become a compromisefor 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 viewpoint10, 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-arccams as an extension of a formulation for two circular-arc cams that has been presented in aprevious paper 12.2. An analytical model for three circular-arc camsAn analytical formulation can be proposed for three circular-arc cams in agreement with designparameters of the model shown in Figs. 1 and 2.Significant parameters for a mechanical design of a three circular-arc cam are: Fig. 1 8; the riseangle as, the dwell angle ar, the return angle ad, the action angle aa as ar ad, the maximumlift h1.Fig. 1. Design parameters for general three circular-arc cams.916C. Lanni et al. / Mechanism and Machine Theory 37 (2002) 915924The characteristic loci of a three circular-arc cams are shown in Fig. 2 as: the first circle C1ofthe cam profile with q1radius and centre C1; the second circle C2of the cam profile with q2radiusand centre C2; the third circle C3of the cam profile with q3radius and centre C3; the base circle C4with radius r and the centre is O; the lift circle C5of the cam profile with (r h1) radius and centreO; the roller circle with radius q centred on the follower axis. In addition significant points are:D ? xD;yD which is the point joining C1with C5; F ? xF;yF which is the point joining C1withC3; G ? xG;yG which is the point joining C3with C2; A ? xA;yA) which is the point joining C2with C4. x and y are Cartesian co-ordinates of points with respect to the fixed frame OXY, whoseorigin O is a point of the cam rotation axis. Additional significant loci are: t13which is the co-incident tangential vector between C1and C3; t15which is the coincident tangential vector betweenC1and C5; t23which is the coincident tangential vector between C2and C3; t24which is the co-incident tangential vector between C2and C4.The model shown in Figs. 1 and 2 can be used to deduce a formulation, which can be usefulboth for characterizing and designing three circular-arc cams. Analytical description can beproposed when the circles are formulated in the suitable form: circle C1with radius q21 x1? xF2 y1? yF2passing through point F asx2 y2? 2xx1? 2yy1? x2F? y2F 2x1xF 2y1yF 01 circle C2with radius q22 x2? xA2 y2? yA2passing through point A asx2 y2? 2xx2? 2yy2? x2A? y2A 2x2xA 2y2yA 02 circle C2with radius q22 x2? xG2 y2? yG2passing through point G asx2 y2? 2xx2? 2yy2? x2G? y2G 2x2xG 2y2yG 03Fig. 2. Characteristic loci for three circular-arc cams.C. Lanni et al. / Mechanism and Machine Theory 37 (2002) 915924917 circle C3with radius q23 x3? xF2 y3? yF2passing through point F asx2 y2? 2xx3? 2yy3? x2F? y2F 2x3xF 2y3yF 04 circle C3with radius q23 x3? xG2 y3? yG2passing through point G asx2 y2? 2xx3? 2yy3? x2G? y2G 2x3xG 2y3yG 05 circle C4with radius r asx2 y2 r26 circle C5with radius (r h1) asx2 y2 r h127Additional characteristic conditions can be expressed in the form as thefirstcircleC1andliftcircleC5musthavethesametangentialvectort15atpointDexpressedasxx1 yy1? x1xD? y1yD 08 the base circle C4and second circle C2must have the same tangential vector t24at point A ex-pressed asxx2 yy2? x2xA? y2yA 09 the second circle C2and third circle C3must have the same tangential vector t23at point G ex-pressed asxx3? x2 yy3? y2 x3xG y3yG? x1xG? y1yG 010 the first circle C1and the second circle C2must have the same tangential vector t12at point Fexpressed asxx1? x3 yy1? y3 x3xF y3yF? x1xF? y1yF 011Eqs. (1)(11) may describe a general model for three circular-arc cams and can be used to drawthe mechanical design as shown in Fig. 2.3. An analytical design procedureEqs. (1)(11) can be used to deduce a suitable system of equations, which allows solving the co-ordinates of the points C1, C2, C3, F and G when suitable data are assumed.It is possible to distinguish four different design cases by using the proposed analytical de-scription.In a first case we can consider that the numeric value of the parameters h1, r, as, ar, ad, q1, q2,and co-ordinates of the points A, C1, C2, D and G are given, and the co-ordinates of points C3, Fare the unknowns. When the action angle aais equal to 180?, the co-ordinate xAof point A is equalto zero. Since A is the point joining C2and C4then the centre C2of the second circle C2lies on theY axis and therefore the co-ordinate x2of the centre C2is equal to zero. By using Eqs. (1)(11) it is918C. Lanni et al. / Mechanism and Machine Theory 37 (2002) 915924possible to deduce a suitable system of equations which allows to solve the co-ordinates of thepoints C3and F. Analytical formulation can be expressed by means of the following conditions: the first circle C1passing across points F and D in the formxF? x12 yF? y12 xD? x12 yD? y1212 the third circle C3passing across points F and G in the formxF? x32 yF? y32 xG? x32 yG? y3213 coincident tangents to C1and C3at the point F in the formx3? x1y3? y1xF? x3yF? y314 coincident tangents to C2and C3at the point G in the formx2? x3y2? y3xG? x2yG? y215When x2 xA 0 are assumed, Eqs. (12)(15) can be expressed asx2F y2F? 2x1xF? 2y1yF? x2D? y2D 2x1xD 2y1yD 0 x2F y2F? 2x3xF? 2y3yF? x2G? y2G 2x3xG 2y3yG 0 xF? x3y3? y1 ? x3? x1yF? y3 0 xGy2? y3 ? x3yG? y2 016If the position of the centre C2is unknown and the direction of the centre C1lies on the ODstraight line, we can approach referring to Fig. 2 a second problem: namely the value of theparameters h1, r, as, ar, ad, q1, and the co-ordinates of the points C2, A, D and G are known andthe co-ordinates of the points C1, F and C3are unknown. Again we may assume aa 180? andconsequently xA x2 0. Two additional conditions are necessary to have a solvable systemtogether with Eq. (9). They are the second circle C2passing across points G and A in the formxG? x22 yG? y22 xA? x22 yA? y2217 straight-line containing points O, A and C2in the formx2yA? xAy2 018Thus, the second case can be solved by Eqs. (16)(18).If the position of the centre C1is unknown but we know that it lies on the OD straight line, wecan approach a third design problem: namely the value of the parameters h1, r, as, ar, ad, q1, andthe co-ordinates of the points A, D and G are known and the co-ordinates of the points C1, C2, Fand C3are unknown. Again we may assume aa 180? and consequently xA x2 0. Two ad-ditional conditions are necessary to have a solvable system together with Eqs. (16)(18). They areC. Lanni et al. / Mechanism and Machine Theory 37 (2002) 915924919 the first circle C1passing across point D in the formxD? x12 yD? y12 q2119 straight-line containing points O, D and C1in the formxDy1? x1yD 020Finally we may approach the fourth case when aa 180? and xA6 0 and also x26 0. Referringto Fig. 1, in which aais the angle between the general position of the point A and the Y axis, thevalue of the parameters h1, r, as, ar, ad, q1, and the co-ordinates of the points A, D and G areknown and the co-ordinates of the points C1, C2, C3and F are unknown. The fourth of Eq. (16)can be expressed asx2? x3yG? y2 ? y2? y3xG? x2 021Thus, 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 tocompute the design parameters. In particular, the proposed formulation has been useful for adesign procedure which makes use of MAPLE to solve for the design unknowns.4. Numerical examplesSeveral numeric examples have been successfully computed in order to prove the soundness andnumerical efficiency of the proposed design formulation. It has been found that only one solutioncan represent a significant circular-arc cam design for any of the formulated design cases.In the Example 1 of Fig. 3 referring to the first design case, the data are given as h1 15 mm,r 40 mm, ar 40?, as ad 70?, q1 17 mm, A ? 0;40 mm, D ? 51:68 mm; 18:81 mmC1? 35:71 mm; 13:00 mm, C2? 0 mm; ?75:64 mm and G ? 22:24 mm; 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 first solution of the analytical formulation. We can note that points F, C1and C3arealigned in the order F, C1and C3and points G, C3and C2in the order G, C3and C2respectively tothe first and second arcs cam profile. Fig. 3(b) shows the second solution of the analytical for-mulation. A cam profile cannot be identified since F point does not lie also on circle C1. Significantpoints F, C1and C3are aligned in the same order with respect to the case in Fig. 3(a); points G, C2and C3are aligned in the C2, G and C3sequential order which is different respect to the case in Fig.3(a) and do not give a cam profile. Fig. 3(c) shows the third solution of analytical formulation thatis similar to the case of Fig. 3(b). Fig. 3(d) shows the fourth solution of analytical formulation. Wecan note that in correspondence of point D there is a cusp. In addition, points F and G are verynear to centre C3so that a sudden change of curvature is obtained in the cam profile as shown inFig. 3(d). Thus a practical feasible design is represented only by Fig. 3(a) that can be characterisedby the proper order F, C1and C3and G, C3and C2of the meaningful points.The feasible numerical solution in Fig. 3(a) is characterised by the values: xF 46:78 mm,yF 25:91 mm, x3 11:99 mm, y3 ?14:47 mm.920C. Lanni et al. / Mechanism and Machine Theory 37 (2002) 915924In the Example 2 of Fig. 3 the data are given as h1 15 mm, r 40 mm, ar 40?,as ad 70?, q1 17 mm, A ? 0; 40 mm, D ? 51:68 mm; 18:81 mm, C1? 35:71 mm;13:00 mm and G ? 22:24 mm; 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: xF 46:78 mm, yF 25:91 mm,x3 11:99 mm, y3 ?14:47 mm, x2 0 mm, y2 ?75:64 mm.In the Example 3 of Fig. 4 referring to the third design case the data are given as h1 15 mm,r 40 mm, ar 40?, as ad 70?, q1 17 mm, A ? 0; 40 mm, D ? 51:68 mm; 18:81 mmand G ? 22:24 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 first solution of analytical formulation. This case is similar to the solution representedin Fig. 3(d). Fig. 4(b) shows the second solution of analytical formulation. We can note that pointF is located below point D so that points F, C1and C3are not aligned. Fig. 3(c) shows the thirdFig. 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) 915924921solution of analytical formulation, which is the same of the case reported in Fig. 3(a). Thus apractical feasible design is represented only by Fig. 4(c).The feasible numerical solution is characterised by the values: xF 46:78 mm, yF 25:91 mm,x3 11:99 mm, y3 ?14:47 mm, x2 0 mm, y2 ?75:64 mm, x1 35:71 mm, y1 13:00 mm.In the Example 4 of Fig. 5 referring to the fourth design case, the data are given as h1 15 mm,r 40 mm, ar 40?, as ad 70?, q1 17 mm, A ? 3:48 mm; 39.84 mm), D ? 51:68 mm;18.81 mm) and G ? 22:24 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 first solution of the analytical formulation. This design is similar to the case reported inFig. 4(a), but the location of point C1is different. Points F, C1and C3are aligned in the C3, F andC1order. Fig. 5(b) shows the second solution of analytical formulation, which is similar to thecase in Fig. 4(a). Fig. 5(c) shows the third solution of analytical formulation. This case shows aFig. 4. Example 3: graphical representation of design solutions for Eqs. (16)(20). Only case (c) is a practical feasibledesign.922C. Lanni et al. / Mechanism and Machine Theory 37 (2002) 915924solution, which is similar to the case reported in Fig. 4(c). Thus a practical feasible design isrepresented only by Fig. 5(c).The feasible numerical solution is characterised by the values: xF 48:15 mm, yF 24:58 mm,x3 16:92 mm, y3 ?4:50 mm, x2 ?40:01 mm, y2 ?457:26 mm, x1 35:71 mm, y1 13:00mm.5. ApplicationsA novel interest can be addressed to approximate design of cam profiles for both new designpurposes and manufacturing needs.Analytical design formulation is required to obtain efficient design algorithms. In addition,closed-form formulation can be also useful to characterise cam profiles in both analysis proce-dures and synthesis criteria. The approximated profiles with circular-arcs can be of particularinterest also to obtain analytical expressions for kinematic characteristics of any profiles that canbe 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 feasibledesign.C. Lanni et al. / Mechanism and Machine Theory 37 (2002) 915924923Indeed, the circular-arc cam profiles have become of current interest because of applications inmini-mechanisms and micro-mechanisms. In fact, when the size of a mechanical design is reducedto the scale of millimeters (mini-mechanisms) and even micron (micro-mechanisms) the manu-facturing of polynomial cam profile becomes difficult and even more complicated is a way toverify it. Therefore, it can be convenient to design circular-arc cam profiles that can be also easilytested experimentally.In addition, stronger and stronger demand of low-cost automation is giving new interest toapproximate designs, which can be used only for specific tasks. This is the case of circular-arc camprofiles that can be conveniently used in low speed machinery or in low-precision applications.6. ConclusionsIn this paper we have proposed an analytical formulation which describes the basic designcharacteristics of three circular-arc cams. A design algorithm has been deduced from the for-mulation, which solves design problems with great numerical efficiency. Numerical examples havebeen reported in the paper to show and discuss the multiple design solutions and the engineeringfeasibility of three circular-arc cams.References1 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 theLast 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 theInternational Conference on Mechanical Transmissions and Mechanisms, Tainjin, 1997, pp. 237238.11 J. Oderfeld, A. Pogorzelski, On designing plane cam mechanisms, in: Proceedings of the Eighth World Congress onthe 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 theFourth Iberoamerican Congress on Mechanical Engineering, Santiago de Chile, vol. 2, 1999.924C. Lanni et al. / Mechanism and Machine Theory 37 (2002) 915924