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畢業(yè)設(shè)計(jì)-翻譯文
三段式圓弧凸輪的解析設(shè)計(jì)（譯）
摘要：
本文對(duì)三段式圓弧凸輪輪廓進(jìn)行了理論性描述。提出了凸輪輪廓的解析式并為以之為尺寸參數(shù)討論。例舉了一些數(shù)值樣例來(lái)證明本理論描述的正確性并表明恰當(dāng)?shù)娜问綀A弧凸輪在工程上是可行的。
1. 序言
凸輪是一種通過(guò)與從動(dòng)件的直接表面接觸來(lái)傳輸預(yù)定運(yùn)動(dòng)的機(jī)構(gòu)。
一般地，從運(yùn)動(dòng)學(xué)[1,2]:來(lái)看，凸輪機(jī)構(gòu)由三部分組成：凸輪（主動(dòng)件）；從動(dòng)件；機(jī)架。凸輪機(jī)構(gòu)廣泛用于現(xiàn)代機(jī)械中，特別是一些自動(dòng)化機(jī)械裝備,內(nèi)燃機(jī)與控制系統(tǒng)[3]。
凸輪機(jī)構(gòu)簡(jiǎn)單而便宜，運(yùn)動(dòng)部件少而且結(jié)構(gòu)緊湊。
凸輪輪廓設(shè)計(jì)主要基于簡(jiǎn)單的幾何曲線，比如:拋物線，諧函數(shù)曲線，擺線，梯形曲線[2,5]以及它們的復(fù)合曲線[1,2,6,7]。
本文主要致力于基于圓弧輪廓的凸輪，即所謂圓弧凸輪。
圓弧凸輪制造容易，用于低速機(jī)構(gòu)中，也可用于微機(jī)械與納米機(jī)械中，因?yàn)榫芗庸た梢酝ㄟ^(guò)利用初等幾何學(xué)準(zhǔn)確地達(dá)到。
這種凸輪的缺點(diǎn)是：凸輪輪廓上不同半徑圓弧交接處會(huì)產(chǎn)生加速度的劇變。[5]
因?yàn)橥ǔＶ挥杏邢迶?shù)量的圓弧，所以其設(shè)計(jì)，制造以及運(yùn)動(dòng)傳輸都不是很復(fù)雜，從而它成為經(jīng)濟(jì)與簡(jiǎn)單的方案，這正是圓弧凸輪[5,8]的優(yōu)點(diǎn)[8]所在。
最近，出于設(shè)計(jì)目的，有人開(kāi)始用描述性視圖給予圓弧凸輪注意。
本文通過(guò)討論其幾何設(shè)計(jì)參量描述了三段式圓弧凸輪。我們?yōu)槿⊥馆喬岢隽私馕鍪阶鳛閷?duì)以前文獻(xiàn)[12]中二弧凸輪解析式的擴(kuò)充。
2. 三段式圓弧凸輪的解析模型
三段式圓弧凸輪解析式中設(shè)計(jì)參量由圖1[8]，圖2給出。
三段式圓弧凸輪設(shè)計(jì)重要參量：圖1：推程運(yùn)動(dòng)角，休止角，回程運(yùn)動(dòng)角，動(dòng)程角，最大舉升位移。
圖1：普通三弧凸輪設(shè)計(jì)參量
圖2：三弧凸輪特征軌跡
三段式圓弧凸輪特征軌跡如圖2所示：由凸輪上半徑ρ1 輪廓形成的第一圓Г1，以及圓心 C1；由凸輪上半徑ρ2 輪廓形成的第二圓Г2，以及圓心 C2；由凸輪上半徑ρ3 輪廓形成的第三圓Г3，以及圓心 C3；由凸輪上半徑r輪廓形成的基圓Г4，以及圓心 O；由凸輪上半徑(r+h1)形成的舉升圓Г5，以及圓心 O；半徑的滾子圓，圓心定于從動(dòng)件軸上。另外，重要的點(diǎn)有：D (,),C1和C5交匯點(diǎn)； F (,) ，C1 和C3交匯點(diǎn)； G (,),C3 和C2交匯點(diǎn)；A (,)，C2和C4交匯點(diǎn)。x 和 y 是與機(jī)架OXY坐標(biāo)系相關(guān)的笛卡爾坐標(biāo)，機(jī)架原點(diǎn)就是凸輪轉(zhuǎn)軸。其他重要軌跡: t13 ，C1 和C3的公切線；t15 ，C1 和 C5的公切線；t23， C2 和 C3的公切線；t24 ，C2 和C4的公切線。
由圖1與圖2可以得出式子，這對(duì)于表現(xiàn)并設(shè)計(jì)三段式圓弧凸輪很有用處。當(dāng)這些圓被以恰當(dāng)?shù)男问奖磉_(dá)時(shí)，解析描述即可得出：
?半徑滿足的圓 C1通過(guò)F點(diǎn)時(shí)滿足：
（1）
?半徑滿足的圓 C2通過(guò)A點(diǎn)時(shí)滿足：
（2）
?半徑滿足的圓 C3通過(guò)G點(diǎn)時(shí)滿足：
（3）
?半徑滿足的圓 C4通過(guò)F點(diǎn)時(shí)滿足：
（4）
?半徑滿足的圓 C5通過(guò)G點(diǎn)時(shí)滿足：
（5）
?半徑r 的圓 C4滿足
（6）
?半徑的圓 C5 滿足
（7）
其他特殊情況可以表示如下：
? 圓 C1 與圓 C5在D點(diǎn)有公切線滿足：
? 基圓 C4 與圓 C2在D點(diǎn)有公切線滿足：
? 圓 C2 與圓 C3在D點(diǎn)有公切線滿足：
? 圓 C1 與圓 C2在D點(diǎn)有公切線滿足：
由式(1)–(11) 可以得到關(guān)于三段式圓弧凸輪的描述并可用于畫出圖2所示的設(shè)計(jì)。
3．解析設(shè)計(jì)過(guò)程
由式(1)–(11) 可以推出一系列等式，當(dāng)C1, C2, C3, F 和 G被賦予合適的值時(shí) ，相關(guān)坐標(biāo)即可得出。
這樣就可以根據(jù)所舉解析描述來(lái)區(qū)分4個(gè)不同的設(shè)計(jì)情況。
第一種情況我們假設(shè)參數(shù)以及A,C1,C2, D和G的坐標(biāo)已知，而點(diǎn)C3, F 坐標(biāo)未知。當(dāng)運(yùn)動(dòng)角 時(shí)，A點(diǎn)橫坐標(biāo)為0 。由于A點(diǎn)是圓C2和C4的交匯點(diǎn)，故C2圓心處于Y軸上，從而C2圓心橫坐標(biāo)也為0。由等式(1)–(11) 可得關(guān)于C3 和 F坐標(biāo)的一系列方程。解析程式表示如下：
? 通過(guò)點(diǎn)F和D的圓 C1表達(dá)式：
? 通過(guò)點(diǎn)F和G的圓 C3表達(dá)式：
?圓C1和圓C3在F點(diǎn)公切線表達(dá)式：
?圓C2和圓C3在G點(diǎn)公切線表達(dá)式：
若，則等式(12)–(15) 可表示為：
（16）
若圓心 C2 未知圓心C1位于直線OD上，我們參考圖2得到第二個(gè)問(wèn)題：即參量 以及點(diǎn) C2, A, D 和G坐標(biāo)均已知，而點(diǎn)C1, F 和 C3 未知。并再設(shè)，而且由上已知，與式(9)聯(lián)立可以得到另外2方程：
? 通過(guò)點(diǎn)G和A的圓 C2表達(dá)式：
? 通過(guò)點(diǎn)O和A的圓心 C2的直線的表達(dá)式：
由等式（17），（18）可解決第2種情況。
若圓心C1 處于直線OD上某處，這便是第3種情況:即參量 以及A, D 和G點(diǎn)坐標(biāo)已知。點(diǎn) C1, C2, F 和 C3 未知。。并再設(shè)，而且由上已知，與式(16)–(18)聯(lián)立可以得到另外2方程：
? 過(guò)點(diǎn)D的圓C1滿足方程：
（19）
? 過(guò)點(diǎn) O, D 和 C1 三點(diǎn)直線滿足：
最后我們得到第4種情況：即當(dāng), ,并且 。圖1中角 間于點(diǎn) A 與 Y 軸。 參量以及點(diǎn)A, D 和 G 坐標(biāo)已知，點(diǎn) C1, C2, C3 和 F 未知。方程組(16)第4式可表示為：
（21）
綜上，三段式圓弧凸輪的一般設(shè)計(jì)可由式 (12)–(14)與(17)–(21) 得到解決。一般的設(shè)計(jì)過(guò)程中的參量計(jì)算?？捎缮厦娴哪Ｊ降玫健＿@一模式在運(yùn)用MAPLE解決未知設(shè)計(jì)量時(shí)優(yōu)勢(shì)更是明顯。
4.數(shù)字樣例
一些數(shù)字樣例的計(jì)算有力地證明了上文模式的正確性與高效率。只有一個(gè)方法可以代表固定程式的圓弧凸輪設(shè)計(jì)。
以圖3中例1作為設(shè)計(jì)樣例1。數(shù)據(jù)如下：
圖三顯示了由等式(16)得出的設(shè)計(jì)結(jié)果。特別的，圖3(a)顯示的是解析式第一種解決方式的結(jié)果：應(yīng)注意到，對(duì)應(yīng)于凸輪輪廓第一，第二圓弧，點(diǎn) F, C1 和 C3 按 F, C1 和 C3 的順序排列，而點(diǎn) G, C3 和 C2 按 G, C3 和 C2 的順序排列。圖3(b)顯示了解析式第二種解決方式的結(jié)果。凸輪輪廓無(wú)法辨別，點(diǎn)F也不在圓上。重要點(diǎn)F, C1 和 C3 按圖3(a)相同順序排列；而點(diǎn) G, C2 和 C3 是按照 C2, G 和 C3 的順序排列這與圖3(a)不同，并且也沒(méi)有給出凸輪輪廓。圖3(c)顯示了解析式第三種解決方式，類似于圖 3(b)。圖 3(d) 顯示了解析式第三種解決方式。我們注意到D點(diǎn)對(duì)應(yīng)一尖點(diǎn)，另外點(diǎn) F 和 G與圓心 C3 靠得很近，所以正如圖3(d)所示，該處曲率變化特別大。故僅有圖3(a)的方案是切實(shí)可行的。各點(diǎn)次序應(yīng)為 F, C1 ，C3 和 G, C3 ， C2 相應(yīng)點(diǎn)。
圖3--例1與例2：方程(16)與方程(16)–(18)設(shè)計(jì)方案的圖示僅(a) 為可行方案。
圖 3(a)方案由以下值確定：
圖3例2，數(shù)據(jù)如下：
其中圖 3 表示的也是由方程(16)–(18)得到的第2方案?？尚袛?shù)字方案取值如下
在圖4例3中，由設(shè)計(jì)情況3，數(shù)據(jù)給定如下：
圖4展示了由方程 (16)–(20)得到的方案。圖4(a)展示的是第一方案結(jié)果，類似于圖3(d)，圖4(b) 展示了解析式第二種解決方案。我們注意到點(diǎn) F 位于點(diǎn) D 下方，故點(diǎn) F, C1 ， C3 不可排列。 圖4(c)展示的于圖3(a)一樣，也是解析式的第3方案。
圖4例3: 方程組(16)–(20)方案的圖形展示。僅圖(c)方案 可行
從而僅有圖4(c)方案可行。可行數(shù)字方案由以下值限定：
在圖5例4中，由第四設(shè)計(jì)方案，可將數(shù)據(jù)給定如下：
圖5展示了由方程組 (16)–(21)得到的方案。圖5(a)展示了第一方案。類似于圖4(a), 但是點(diǎn)C1方位有異。 點(diǎn) F, C1 和 C3 以 C3, F 和 C1 的順序排列。圖5(b) 展示了解析式第二方案，類似于圖4(a)。圖5(c)展示了解析式第三方案，類似于圖4(c)。
圖5例4:方程組(16)–(21)所得方案圖示.僅方案(c) 可行
從而可得可行方案為圖5(c)中方案?？尚袛?shù)字方案之賦值：
5. 應(yīng)用
本文旨在提出凸輪輪廓近似設(shè)計(jì)新的設(shè)計(jì)途徑并滿足其制造需求。
由設(shè)計(jì)解析式可以獲得高效率的設(shè)計(jì)運(yùn)算法則。緊湊的解析式更可以在凸輪的分析過(guò)程及其綜合特性的實(shí)現(xiàn)中發(fā)揮作用。由圓弧組成的近似輪廓，在取得任何含近似圓弧輪廓的動(dòng)力學(xué)特性的分析表達(dá)式具有特殊的重要性。
的確，由于在小型及微型機(jī)械中的應(yīng)用，圓弧形凸輪輪廓已經(jīng)具有了相當(dāng)?shù)闹匾?。事?shí)上，當(dāng)構(gòu)造設(shè)計(jì)已經(jīng)提升到毫微米級(jí)別的時(shí)候，多項(xiàng)式曲線輪廓的凸輪的制造變得相當(dāng)困難，要想校驗(yàn)更如登天。因此，設(shè)計(jì)便利的圓弧輪廓凸輪成為首選，而其實(shí)驗(yàn)性測(cè)試也是方便。
另外，對(duì)低成本自動(dòng)化與日俱增的需求，也賦予這些僅適于特殊用途的近似設(shè)計(jì)新的重要性。圓弧凸輪輪廓方案可以方便地用于低速或低精度機(jī)械中。
6. 綜述
本文提出了有關(guān)三段式圓弧凸輪輪廓基本設(shè)計(jì)的解析方法。從該法我們推導(dǎo)出了１個(gè)設(shè)計(jì)算法，從而可以高效地解決該方向一些設(shè)計(jì)問(wèn)題。另外還舉出了一些數(shù)字樣例以展示與討論三段式圓弧凸輪的多重設(shè)計(jì)以及工程可行性問(wèn)題。
７．參考文獻(xiàn)
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9
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