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汽车变速箱三维设计与仿真【三维PROE+仿真动画】[CAD图纸和文档资料]

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设计题目：汽车变速箱三维设计与仿真


  【摘要】

　　　变速器用来改变发动机传到驱动轮上的转矩和转速，目的是在原地起步，爬坡，转弯，加速等各种行驶工况下，使汽车获得不同的牵引力和速度，同时使发动机在最有利工况范围内工作。变速器设有空挡和倒挡。需要时变速器还有动力输出功能。

　　　因为变速箱在低档工作时作用有较大的力，所以一般变速箱的低档都布置靠近轴的后支承处，然后按照从低档到高档顺序布置各档位齿轮。这样做既能使轴有足够大的刚性，又能保证装配容易。变速箱整体结构刚性与轴和壳体的结构有关系。一般通过控制轴的长度即控制档数，来保证变速箱有足够的刚性。

　　　本文设计研究了三轴式五挡手动变速器，对变速器的工作原理做了阐述，变速器的各挡齿轮和轴做了详细的设计计算，并进行了强度校核，对一些标准件进行了选型。变速器的传动方案设计。简单讲述了变速器中各部件材料的选择。

　　【关键词】挡数；传动比；齿数；轴

　　【Abstract】

　　　Transmission to change the engine reached on the driving wheel torque and speed, is aimed at marking start, climbing, turning, accelerate various driving conditions, the car was different traction and speed Meanwhile engine in the most favorable working conditions within the scope of the work. And the trans mission in neutral gear with reverse gear. Transmission also need power output function.

　　　Gearbox because of the low-grade work at a larger role, In general, the low-grade gearbox layout are close to the axis after support, Following from low-grade to high-grade order of the layout of stalls gear. This will not only allow axis are large enough for a rigid, but also ensures easy assembly. Gear box overall structure and rigid axle and the shell structure of relations. Generally through the control shaft length control over several stalls to ensure that adequate gear box rigid. 

　　　This paper describes the design of three-axis five block manual trans mission, the transmission principle of work elaborated, Transmission of the gear shaft and do a detailed design, and the intensity of a school. For some standard parts for the selection. Transmission Trans mission program design. A brief description of the trans mission of all components of the material choice. 

　　[Keywords]: block; Transmission ratio; Teeth; Axis 


目  录

引言 1

1 总体方案设计 3

1.1 汽车参数的选择 3

1．2变速器设计应满足的基本要求 3

2 变速器传动机构布置方案 4

2.1 传动机构布置方案分析 4

2.1.1 固定轴式变速器 4

2.1.2 倒挡布置方案 6

2.1.3 其他问题 7

3 零部件结构方案分析 9

3.1 齿轮形式 9

3.2 换挡机构形式 9

3.3 变速器轴承 11

4变速器设计和计算 12

4.1 挡数 12

4.2 传动比范围 12

4.3 中心距A 12

4.4 外形尺寸 13

4.5 轴的直径 13

4.6 齿轮参数 13

4.6.1 模数的选取 13

4.6.2 压力角 14

4.6.3 螺旋角 14

4.6.4 齿宽b 16

4.6.5 变位系数的选择原则 16

4.7 各挡齿轮齿数的分配 17

4.7.1 确定一挡齿轮的齿数 17

4.7.2 对中心距进行修正 18

4.7.3 确定常啮合传动齿轮副的齿数 18

4.7.4 确定其他各挡的齿数 19

4.7.5 确定倒挡齿轮齿数 20

5 变速器的校核 21

5.1 齿轮的损坏形式 21

5.2 齿轮强度计算 21

5.2.1 齿轮弯曲强度计算 21

5.2.2 轮齿接触应力计算 22

6 同步器的选型 24

6.1 锁销式同步器 24

6.1.1 锁销式同步器结构 24

6.1.2 锁销式同步器工作原理 25

6.2 锁环式同步器 26

6.2.1 锁环式同步器结构 26

6.2.2 锁环式同步器工作原理 26

6.2.3 锁环式同步器主要尺寸的确定 27

7 变速器操纵机构 30

7.1 直接操纵手动换挡变速器 30

7.2 远距离操纵手动换挡变速器 31

8 三维建模 32

8.1各档位齿轮的建模 32

8.2箱体和箱体上盖的建模见下图9.11和9.12 37

8.3汽车变速箱整体模型和爆炸图见下图9.13和9.14 38

9 变速箱的仿真 40

9.1 方针的的第一步，完成三维变速箱的装配 40

9.1.1新建装配环境 40

9.1.2装配倒档轴组件 40

9.1.3装配中间轴组件 41

9.1.4装配一二轴组件 41

9.1.5完成变速箱的全部装配 42

9.2进行仿真 42

9.2.1定义齿轮副 43

9.2.2定义伺服电动机 43

9.2.3机构分析 43

9.3保存仿真动画 43

10 结论 44

致谢 45

参考文献 46

附录 47


引言

　　　现代汽车的动力装置，几乎都采用往复活塞式内燃机。它具有相当多的优点，如体积小，质量轻，工作可靠，使用方便等。但其性能与汽车的动力性和经济性之间存在着较大的矛盾。如在坡道上行驶时，所需的牵引力往往是发动机所能提供的牵引力的数倍。而且一般发动机如果直接与车轮相连，其输出转速换算到对应的汽车车速上，将达到现代汽车极限速度的数倍。上述发动机牵引力、转速与汽车牵引力、车速要求之间的矛盾，单靠现代汽车内燃机本身是无法解决的。因此就出现了车用变速箱和主减速器。它们的共同努力使驱动轮的扭矩增大到发动机扭矩的若干倍，同时又可使其转速减小到发动机转速的几分之一。

　　　另外，现代汽车的使用条件极为复杂，在不同场合下有不同的要求。往往要受到如载运量、道路坡度、路面好坏及交通是否通畅等条件的影响。这就要求汽车的牵引力和车速能在较大范围内变化，以适应使用的要求。在条件良好的平直路面上要能以高速行驶，而在路面不平和有较大坡度时能提供较大的扭矩。变速箱的多挡位选择就能满足这些需求。此外，发动机在不同工况下，燃油的消耗量也是不一样的。驾驶员可以根据具体情况，选择变速箱的某一挡位，来减少燃油的消耗。在某些情况下，汽车还需要能倒向行驶。发动机本身是不可能倒转的，只有靠变速箱的倒挡齿轮来实现。


内容简介：
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18，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1 轮廓形成的第一圆1，以及圆心 C1；由凸轮上半径2 轮廓形成的第二圆2，以及圆心 C2；由凸轮上半径3 轮廓形成的第三圆3，以及圆心 C3；由凸轮上半径r轮廓形成的基圆4，以及圆心 O；由凸轮上半径(r+h1)形成的举升圆5，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。参考文献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 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, 1999.924 C. Lanni et al. / Mechanism and Machine Theory 37 (2002) 9159249An analytical design for three circular-arc cams Chiara Lanni, Marco Ceccarelli *, Giorgio Figliolini Dipartimento di Meccanica, Strutture, Ambiente e Territorio, Universit? a a 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 profi les. An ana- lytical formulation for cam profi les 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. ? 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 diff erent 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 fi 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 profi le 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 profi les, 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 ? 2002 Elsevier Science Ltd. All rights reserved. PII: S0094-114X(02)00032-0 Mechanism and Machine Theory 37 (2002) 915924 /locate/mechmt 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 profi le points where arcs of diff erent 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. An analytical model for three circular-arc cams An analytical formulation can be proposed for three circular-arc cams in agreement with design parameters of the model shown in Figs. 1 and 2. Signifi cant parameters for a mechanical design of a three circular-arc cam are: Fig. 1 8; the rise angle as, the dwell angle ar, the return angle ad, the action angle aa as ar ad, the maximum lift h1. Fig. 1. Design parameters for general three circular-arc cams. 916C. Lanni et al. / Mechanism and Machine Theory 37 (2002) 915924 The characteristic loci of a three circular-arc cams are shown in Fig. 2 as: the fi rst circle C1of the cam profi le with q1radius and centre C1; the second circle C2 of the cam profi le with q2radius and centre C2; the third circle C3 of the cam profi le with q3radius and centre C3; the base circle C4 with radius r and the centre is O; the lift circle C5 of the cam profi le with (r h1) radius and centre O; the roller circle with radius q centred on the follower axis. In addition signifi cant points are: D ? xD;yD which is the point joining C1with C5; F ? xF;yF which is the point joining C1with C3; G ? xG;yG which is the point joining C3with C2; A ? xA;yA) which is the point joining C2 with C4 . x and y are Cartesian co-ordinates of points with respect to the fi xed frame OXY, whose origin O is a point of the cam rotation axis. Additional signifi cant loci are: t13which is the co- incident tangential vector between C1and C3; t15which is the coincident tangential vector between C1and 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 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 C1with radius q2 1 x1? xF2 y1? yF2passing through point F as x2 y2? 2xx1? 2yy1? x2 F ? y2 F 2x1xF 2y1yF 01 circle C2with radius q2 2 x2? xA2 y2? yA2passing through point A as x2 y2? 2xx2? 2yy2? x2 A? y 2 A 2x2xA 2y2yA 02 circle C2with radius q2 2 x2? xG2 y2? yG2passing through point G as x2 y2? 2xx2? 2yy2? x2 G? y 2 G 2x2xG 2y2yG 03 Fig. 2. Characteristic loci for three circular-arc cams. C. Lanni et al. / Mechanism and Machine Theory 37 (2002) 915924917 circle C3with radius q2 3 x3? xF2 y3? yF2passing through point F as x2 y2? 2xx3? 2yy3? x2 F ? y2 F 2x3xF 2y3yF 04 circle C3with radius q2 3 x3? xG2 y3? yG2passing through point G as x2 y2? 2xx3? 2yy3? x2 G? y 2 G 2x3xG 2y3yG 05 circle C4with radius r as x2 y2 r26 circle C5with radius (r h1) as x2 y2 r h127 Additional characteristic conditions can be expressed in the form as thefi rstcircleC1andliftcircleC5musthavethesametangentialvectort15atpointDexpressedas xx1 yy1? x1xD? y1yD 08 the base circle C4and second circle C2must have the same tangential vector t24at point A ex- pressed as xx2 yy2? x2xA? y2yA 09 the second circle C2and third circle C3must have the same tangential vector t23at point G ex- pressed as xx3? x2 yy3? y2 x3xG y3yG? x1xG? y1yG 010 the fi rst circle C1and the second circle C2must have the same tangential vector t12at point F expressed as xx1? x3 yy1? y3 x3xF y3yF? x1xF? y1yF 011 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. An analytical design procedure Eqs. (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 diff erent design cases by using the proposed analytical de- scription. In a fi rst 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, F are the unknowns. When the action angle aais equal to 180?, the co-ordinate xAof point A is equal to zero. Since A is the point joining C2and C4then the centre C2of the second circle C2lies on the Y axis and therefore the co-ordinate x2of the centre C2is equal to zero. By using Eqs. (1)(11) it is 918C. 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 C3and F. Analytical formulation can be expressed by means of the following conditions: the fi rst circle C1passing across points F and D in the form xF? x12 yF? y12 xD? x12 yD? y1212 the third circle C3passing across points F and G in the form xF? x32 yF? y32 xG? x32 yG? y3213 coincident tangents to C1and C3at the point F in the form x3? x1 y3? y1 xF? x3 yF? y3 14 coincident tangents to C2and C3at the point G in the form x2? x3 y2? y3 xG? x2 yG? y2 15 When x2 xA 0 are assumed, Eqs. (12)(15) can be expressed as x2 F y2 F ? 2x1xF? 2y1yF? x2 D? y 2 D 2x1xD 2y1yD 0 x2 F y2 F ? 2x3xF? 2y3yF? x2 G? y 2 G 2x3xG 2y3yG 0 xF? x3y3? y1 ? x3? x1yF? y3 0 xGy2? y3 ? x3yG? y2 0 16 If the position of the centre C2is unknown and the direction of the centre C1lies on the OD straight line, we can approach referring to Fig. 2 a second problem: namely the value of the parameters h1, r, as, ar, ad, q1, and the co-ordinates of the points C2, A, D and G are known and the co-ordinates of the points C1, F and C3are unknown. Again we may assume aa 180? and consequently xA x2 0. Two additional conditions are necessary to have a solvable system together with Eq. (9). They are the second circle C2passing across points G and A in the form xG? x22 yG? y22 xA? x22 yA? y2217 straight-line containing points O, A and C2in the form x2yA? xAy2 018 Thus, 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, we can approach a third design problem: namely the value of the parameters h1, r, as, ar, ad, q1, and the co-ordinates of the points A, D and G are known and the co-ordinates of the points C1, C2, F and 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 are C. Lanni et al. / Mechanism and Machine Theory 37 (2002) 915924919 the fi rst circle C1passing across point D in the form xD? x12 yD? y12 q2 1 19 straight-line containing points O, D and C1in the form xDy1? x1yD 020 Finally we may approach the fourth case when aa 180? and xA6 0 and also x26 0. Referring to Fig. 1, in which aais the angle between the general position of the point A and the Y axis, the value of the parameters h1, r, as, ar, ad, q1, and the co-ordinates of the points A, D and G are known and the co-ordinates of the points C1, C2, C3and F are unknown. The fourth of Eq. (16) can be expressed as x2? x3yG? y2 ? y2? y3xG? x2 021 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. Numerical examples Several numeric examples have been successfully computed in order to prove the soundness and numerical effi ciency of the proposed design formulation. It has been found that only one solution can represent a signifi cant circular-arc cam design for any of the formulated design cases. In the Example 1 of Fig. 3 referring to the fi rst 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 mm C1? 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 fi rst solution of the analytical formulation. We can note that points F, C1and C3are aligned in the order F, C1and C3and points G, C3and C2in the order G, C3and C2respectively to the fi rst and second arcs cam profi le. Fig. 3(b) shows the second solution of the analytical for- mulation. A cam profi le cannot be identifi ed since F point does not lie also on circle C1 . Signifi cant points F, C1and C3are aligned in the same order with respect to the case in Fig. 3(a); points G, C2 and C3are aligned in the C2, G and C3 sequential order which is diff erent respect to the case in Fig. 3(a) and do not give a cam profi le. Fig. 3(c) shows the third solution of analytical formulation that is similar to the case of Fig. 3(b). Fig. 3(d) shows the fourth solution of analytical formulation. We can note that in correspondence of point D there is a cusp. In addition, points F and G are very near to centre C3 so that a sudden change of curvature is obtained in the cam profi le as shown in Fig. 3(d). Thus a practical feasible design is represented only by Fig. 3(a) that can be characterised by 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) 915924 In 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 mm and 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 fi rst solution of analytical formulation. This case is similar to the solution represented in Fig. 3(d). Fig. 4(b) shows the second solution of analytical formulation. We can note that point F is located below point D so that points F, C1and C3are 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) 915924921 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: 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 fi rst solution of the analytical formulation. This design is similar to the case reported in Fig. 4(a), but the location of point C1 is diff erent. Points F, C1and C3are aligned in the C3, F and C1order. 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. 922C. 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: 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 13:00 mm. 5. Applications A novel interest can be addressed to approximate design of cam profi les for both new design purposes and manufacturing needs. Analytical design formulation is required to obtain effi cient design algorithms. In addition, closed-form formulation can be also useful to characterise cam profi les in both analysis proce- dures and synthesis criteria. The approximated profi les with circular-arcs can be of particular interest also to obtain analytical expressions for kinematic characteristics of any profi les 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) 915924923 Indeed, the circular-arc cam profi les 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 profi le becomes diffi cult and even more complicated is a way to verify it. Therefore, it can be convenient to design c
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