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毕业设计-翻译文
三段式圆弧凸轮的解析设计（译）
摘要：
本文对三段式圆弧凸轮轮廓进行了理论性描述。提出了凸轮轮廓的解析式并为以之为尺寸参数讨论。例举了一些数值样例来证明本理论描述的正确性并表明恰当的三段式圆弧凸轮在工程上是可行的。
1. 序言
凸轮是一种通过与从动件的直接表面接触来传输预定运动的机构。
一般地，从运动学[1,2]:来看，凸轮机构由三部分组成：凸轮（主动件）；从动件；机架。凸轮机构广泛用于现代机械中，特别是一些自动化机械装备,内燃机与控制系统[3]。
凸轮机构简单而便宜，运动部件少而且结构紧凑。
凸轮轮廓设计主要基于简单的几何曲线，比如:抛物线，谐函数曲线，摆线，梯形曲线[2,5]以及它们的复合曲线[1,2,6,7]。
本文主要致力于基于圆弧轮廓的凸轮，即所谓圆弧凸轮。
圆弧凸轮制造容易，用于低速机构中，也可用于微机械与纳米机械中，因为精密加工可以通过利用初等几何学准确地达到。
这种凸轮的缺点是：凸轮轮廓上不同半径圆弧交接处会产生加速度的剧变。[5]
因为通常只有有限数量的圆弧，所以其设计，制造以及运动传输都不是很复杂，从而它成为经济与简单的方案，这正是圆弧凸轮[5,8]的优点[8]所在。
最近，出于设计目的，有人开始用描述性视图给予圆弧凸轮注意。
本文通过讨论其几何设计参量描述了三段式圆弧凸轮。我们为三弧凸轮提出了解析式作为对以前文献[12]中二弧凸轮解析式的扩充。
2. 三段式圆弧凸轮的解析模型
三段式圆弧凸轮解析式中设计参量由图1[8]，图2给出。
三段式圆弧凸轮设计重要参量：图1：推程运动角，休止角，回程运动角，动程角，最大举升位移。
图1：普通三弧凸轮设计参量
图2：三弧凸轮特征轨迹
三段式圆弧凸轮特征轨迹如图2所示：由凸轮上半径ρ1 轮廓形成的第一圆Г1，以及圆心 C1；由凸轮上半径ρ2 轮廓形成的第二圆Г2，以及圆心 C2；由凸轮上半径ρ3 轮廓形成的第三圆Г3，以及圆心 C3；由凸轮上半径r轮廓形成的基圆Г4，以及圆心 O；由凸轮上半径(r+h1)形成的举升圆Г5，以及圆心 O；半径的滚子圆，圆心定于从动件轴上。另外，重要的点有：D (,),C1和C5交汇点； F (,) ，C1 和C3交汇点； G (,),C3 和C2交汇点；A (,)，C2和C4交汇点。x 和 y 是与机架OXY坐标系相关的笛卡尔坐标，机架原点就是凸轮转轴。其他重要轨迹: t13 ，C1 和C3的公切线；t15 ，C1 和 C5的公切线；t23， C2 和 C3的公切线；t24 ，C2 和C4的公切线。
由图1与图2可以得出式子，这对于表现并设计三段式圆弧凸轮很有用处。当这些圆被以恰当的形式表达时，解析描述即可得出：
•半径满足的圆 C1通过F点时满足：
（1）
•半径满足的圆 C2通过A点时满足：
（2）
•半径满足的圆 C3通过G点时满足：
（3）
•半径满足的圆 C4通过F点时满足：
（4）
•半径满足的圆 C5通过G点时满足：
（5）
•半径r 的圆 C4满足
（6）
•半径的圆 C5 满足
（7）
其他特殊情况可以表示如下：
• 圆 C1 与圆 C5在D点有公切线满足：
• 基圆 C4 与圆 C2在D点有公切线满足：
• 圆 C2 与圆 C3在D点有公切线满足：
• 圆 C1 与圆 C2在D点有公切线满足：
由式(1)–(11) 可以得到关于三段式圆弧凸轮的描述并可用于画出图2所示的设计。
3．解析设计过程
由式(1)–(11) 可以推出一系列等式，当C1, C2, C3, F 和 G被赋予合适的值时 ，相关坐标即可得出。
这样就可以根据所举解析描述来区分4个不同的设计情况。
第一种情况我们假设参数以及A,C1,C2, D和G的坐标已知，而点C3, F 坐标未知。当运动角 时，A点横坐标为0 。由于A点是圆C2和C4的交汇点，故C2圆心处于Y轴上，从而C2圆心横坐标也为0。由等式(1)–(11) 可得关于C3 和 F坐标的一系列方程。解析程式表示如下：
• 通过点F和D的圆 C1表达式：
• 通过点F和G的圆 C3表达式：
•圆C1和圆C3在F点公切线表达式：
•圆C2和圆C3在G点公切线表达式：
若，则等式(12)–(15) 可表示为：
（16）
若圆心 C2 未知圆心C1位于直线OD上，我们参考图2得到第二个问题：即参量 以及点 C2, A, D 和G坐标均已知，而点C1, F 和 C3 未知。并再设，而且由上已知，与式(9)联立可以得到另外2方程：
• 通过点G和A的圆 C2表达式：
• 通过点O和A的圆心 C2的直线的表达式：
由等式（17），（18）可解决第2种情况。
若圆心C1 处于直线OD上某处，这便是第3种情况:即参量 以及A, D 和G点坐标已知。点 C1, C2, F 和 C3 未知。。并再设，而且由上已知，与式(16)–(18)联立可以得到另外2方程：
• 过点D的圆C1满足方程：
（19）
• 过点 O, D 和 C1 三点直线满足：
最后我们得到第4种情况：即当, ,并且 。图1中角 间于点 A 与 Y 轴。 参量以及点A, D 和 G 坐标已知，点 C1, C2, C3 和 F 未知。方程组(16)第4式可表示为：
（21）
综上，三段式圆弧凸轮的一般设计可由式 (12)–(14)与(17)–(21) 得到解决。一般的设计过程中的参量计算常可由上面的模式得到。这一模式在运用MAPLE解决未知设计量时优势更是明显。
4.数字样例
一些数字样例的计算有力地证明了上文模式的正确性与高效率。只有一个方法可以代表固定程式的圆弧凸轮设计。
以图3中例1作为设计样例1。数据如下：
图三显示了由等式(16)得出的设计结果。特别的，图3(a)显示的是解析式第一种解决方式的结果：应注意到，对应于凸轮轮廓第一，第二圆弧，点 F, C1 和 C3 按 F, C1 和 C3 的顺序排列，而点 G, C3 和 C2 按 G, C3 和 C2 的顺序排列。图3(b)显示了解析式第二种解决方式的结果。凸轮轮廓无法辨别，点F也不在圆上。重要点F, C1 和 C3 按图3(a)相同顺序排列；而点 G, C2 和 C3 是按照 C2, G 和 C3 的顺序排列这与图3(a)不同，并且也没有给出凸轮轮廓。图3(c)显示了解析式第三种解决方式，类似于图 3(b)。图 3(d) 显示了解析式第三种解决方式。我们注意到D点对应一尖点，另外点 F 和 G与圆心 C3 靠得很近，所以正如图3(d)所示，该处曲率变化特别大。故仅有图3(a)的方案是切实可行的。各点次序应为 F, C1 ，C3 和 G, C3 ， C2 相应点。
图3--例1与例2：方程(16)与方程(16)–(18)设计方案的图示仅(a) 为可行方案。
图 3(a)方案由以下值确定：
图3例2，数据如下：
其中图 3 表示的也是由方程(16)–(18)得到的第2方案。可行数字方案取值如下
在图4例3中，由设计情况3，数据给定如下：
图4展示了由方程 (16)–(20)得到的方案。图4(a)展示的是第一方案结果，类似于图3(d)，图4(b) 展示了解析式第二种解决方案。我们注意到点 F 位于点 D 下方，故点 F, C1 ， C3 不可排列。 图4(c)展示的于图3(a)一样，也是解析式的第3方案。
图4例3: 方程组(16)–(20)方案的图形展示。仅图(c)方案 可行
从而仅有图4(c)方案可行。可行数字方案由以下值限定：
在图5例4中，由第四设计方案，可将数据给定如下：
图5展示了由方程组 (16)–(21)得到的方案。图5(a)展示了第一方案。类似于图4(a), 但是点C1方位有异。 点 F, C1 和 C3 以 C3, F 和 C1 的顺序排列。图5(b) 展示了解析式第二方案，类似于图4(a)。图5(c)展示了解析式第三方案，类似于图4(c)。
图5例4:方程组(16)–(21)所得方案图示.仅方案(c) 可行
从而可得可行方案为图5(c)中方案。可行数字方案之赋值：
5. 应用
本文旨在提出凸轮轮廓近似设计新的设计途径并满足其制造需求。
由设计解析式可以获得高效率的设计运算法则。紧凑的解析式更可以在凸轮的分析过程及其综合特性的实现中发挥作用。由圆弧组成的近似轮廓，在取得任何含近似圆弧轮廓的动力学特性的分析表达式具有特殊的重要性。
的确，由于在小型及微型机械中的应用，圆弧形凸轮轮廓已经具有了相当的重要性。事实上，当构造设计已经提升到毫微米级别的时候，多项式曲线轮廓的凸轮的制造变得相当困难，要想校验更如登天。因此，设计便利的圆弧轮廓凸轮成为首选，而其实验性测试也是方便。
另外，对低成本自动化与日俱增的需求，也赋予这些仅适于特殊用途的近似设计新的重要性。圆弧凸轮轮廓方案可以方便地用于低速或低精度机械中。
6. 综述
本文提出了有关三段式圆弧凸轮轮廓基本设计的解析方法。从该法我们推导出了１个设计算法，从而可以高效地解决该方向一些设计问题。另外还举出了一些数字样例以展示与讨论三段式圆弧凸轮的多重设计以及工程可行性问题。
７．参考文献
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924 C. Lanni et al. / Mechanism and Machine Theory 37 (2002) 915–924
9
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
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