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凸轮形状设计 的混合方法和 一般的盘形凸轮轮廓加工的机制 关键字： 盘形凸轮机构 ，凸轮，从动件，外形设计，数控加工，接触点， NC 数据，即时速度中心，双圆弧曲线拟合。 盘形凸轮机构可以很容易的在凸轮和从动件接触的地方产生积极的和功能的运动。一般情况下凸轮机构在很多领域采用的是机械式控制，自动化和工业机械。为了得到从动件的准确运动，凸轮的轮廓的设计和加工必须是精确的。本篇文章提出了一个以即时速度为中心的设计方法和一个为加工四种往复和摆动运动，滚子或者平面从动件类型的盘形凸轮机构的双圆弧拟合方法。本文的关键是引进设计 程序和制造程序相结合的混合动力系统。文章的主要思想是，从设计参数出发用最小的加工数据建立准确的双圆弧曲线的安装。可以直接由凸轮的接触角和设计程序中给出的从动件准确的定义的双圆弧曲线径向方向上的角朝向双圆弧中心。一个的应用程序验证了在给定的加工公差的情况下使用最小的 NC 数据设计凸轮和加工凸轮这个 所提出的方法是正确的。 1.介绍 平板凸轮机构是一种广泛使用的由凸轮和从动件的连续接触的运动所构成的机械部件，可以由凸轮的旋转很容易的让从动件产生任何功能的运动。凸轮机构通过组合不同形状的凸轮和不同运动形态的从动件来拥有 不同类型的机构，如板形或者圆筒形的凸轮，滚子或者平底的从动件，往返或者摆动的运动。 尽管凸轮机构有连接数少，结构简单，正向运动以及紧凑的尺寸这些优点，但是凸轮机构还是要有精确的外形设计和精密的加工程序来满足机械性能要求。在较低水平的设计和制造的情况下，凸轮机构的整体系统会有沉重的振动，噪音，分隔和超载。为了避免这些影响，凸轮机构必须精心，准确的设计和精确的加工。事实上，一种混合的 CAD/CAM 的方法有可能是最好的解决方案，这种解决方案是从设计过程中得到的形状数据直接结合得到加工数据的一种加工工艺。从描述文件 的凸轮的加工数据去加工最常用的是直线插补和圆弧插补。 Shin 等人提出在两个圆弧的连接点因为不连续的曲率半径和不连续的斜率所以直线插补的精度低，圆弧插补不能保持精度。近日，Jung 等人和 Yang 等人提出参数化的差值使用 B 样条和 NURBS 曲线。双圆弧插补被广泛的应用和依赖在方向角朝向中心的双圆弧曲线上。博尔顿描述了一个在两个点的切线角为基础的双圆弧曲线，帕金森以及摩顿在三个点的一元二次方程的基础上得到了双圆弧曲线，米克和沃尔顿用样条曲线的类型来构建双圆弧曲线。 Schonherr 介绍了一种减小双圆弧曲线的半径的方 法。 nts通常，这些插值方法使得加工点增加，然后机构弯曲的形状上过多的加工数据使得加工误差也随之增多。因此，精确的机械加工过程中，在加工误差允许的范围内应该尽量的减少加工点来保持加工精度。本文介绍了 CAD/CAM 的混合方法的 3 个步骤在四个不同类型的板凸轮机构中。首先，凸轮的形状是由在瞬时速度时的约束中心和凸轮跟从动件在接触点的接触角来决定的。第二个步骤是把接触角变换成中心的方向角，然后，计算双圆弧曲线的半径。最后，加工数据最小化是通过膨胀和收缩双圆弧部分，膨胀还是收缩是看凸轮轮廓点是位于加工公差范围之内还是之外 。 2.双圆弧拟合精度的阐明 在常见的盘形凸轮的设计过程中只有凸轮形状的轮廓数据被定义，然后在数控（数据控制）过程中加工数据必须根据任何的曲线拟合来开发。圆形的嵌合，这是最广泛的用于机械加工，具有不可靠性如图 1 所示。由三个点（ P1， p2， p3）所确定的一个圆具有半径 R1，另一个由其他 3 个点（ P2,P3,P4）所确定的圆具有半径 R2。嵌合这两个圆，通过点（ P1,P2,P3,P4） ,但是这些点组成的是不连续的斜坡，而且是不连续的半径在跨距中间。这些缺点使得拟合的曲线精度很低，然后会使得凸轮机构在高速转动的过程中会有 更高的振动。 图 1，圆形拟合上的缺陷 nts 图 2：连续的双圆弧拟合 图 2 显示了一个通过 biarcs 所得的连续的曲线，通过轮廓点（ P1,P2,P3,P4）。在这种情况下双圆弧曲线拥有四个半径。半径 R1 是通过 P1 到 S1， R2 是通过 S1 到S2， R3 是通过 S2 到 S3， R4 是通过 S3 到 P4。每一点上的双圆弧曲线的斜率是连续的和独特的。同样，中间跨距点（ S1， S2， S3）是连续的没有跳跃的半径，这样就双圆弧曲线可以保持较高的精度水平。 如图二所示，双圆弧拟合是高度依赖于径向方向的角度（ ）。常见的凸轮机构设计过程中只定义了配置文件数据，然后圆形的嵌合加工过程中必须使用的角度。这个过程因为不正确的角度使精度较低。但是，本文所提出的方法可以定义正确的角度，这个方法是通过凸轮轮廓的设计过程中直接给出，然后加工数据保持较高的的准确性。 3.盘形凸轮的形状设计 3.1 凸轮机构的位移特性 如图三所示，对于往复运动的滚子从动件的盘形凸轮机构，从动件运动的运动学特性，可以被定义为线性位移 Y，一阶导数 Y ，二阶导数 Y 凸轮的旋转角度 c 。属性是作为角位移以防摆动从动件。本文给出的即时速度的中心方法是使用位移和确定凸轮形状衍生工具。 nts 图 3：往复滚子从动件的凸轮机构 3.2 瞬间速度中心基础上的外形设计 如图 4 所示，点 Q 是一条直线穿过点接触点 C 和滚子中心与一条水平线的交点，它是即时速度的中心。点 Q 的速度与凸轮的旋转 速度成比例在式子（ 1）中，滚子在R 点的速度在式子（ 2）中定义成从动件的线性速度。 通过凸轮机构的运动特性，即时速度中心的速度QV和从动件的速度 RV 相同。因此，速度条件给出了式（ 3）中的即时速度中心的位置。 nts 图 4：凸轮与从动件的接触位置 如图四所示，接触角是有一个滑动速度线和从动件滚子的法线之间的夹角由式子（ 4）定义，接触点坐标由 式子（ 5）给出，滚子中心坐标（yx RR,）可以由位移（ Y）和凸轮机构给定的几何条件（素圆和偏心）计算出来，其中 rR 是滚子的半径。最后，联系点（xC和yC）是由式子（ 5）给出的。 nts 图 5：往复式 平面从动件盘形凸轮 图 5 显示了一个往复平面从动件的凸轮机构。即时速度中心 Q 位于水平线和根据式子（ 6）在速度中心在瞬间的速度条件基础下得到。然后，接触点由式子（ 7）定义。 图 6 是一种摆动滚子从动件的机构，凸轮中心和即时速度中心的距离由方程（ 8）得到。方程（ 9）表示凸轮和滚子之间的接触角，方程（ 10）表示接触点。在这里，xyZ是轴和凸轮中心的距离。 nts 图 6：摆动滚子从动件盘形凸轮 图 7 表示摆动平面从动件的凸轮机 构，由方程（ 11）得到即时速度中心点的位置，由方程（ 12）得到接触点位置。 nts 图 7：摆动平面从动件盘形凸轮 最后，可以由方程（ 13）通过变换接触点和凸轮旋转的反向角来确定凸轮形状的外形。这边ySSx和是凸轮的轮廓坐标。 3.3 接触点内部正常的角度 图 4-7 显示了，在四种不同情况下的凸轮机构在每一个接触点的法线。一个内部的正常角（ ）在本文中 被定义为接触点连接到凸轮中心和即时速度中心的线之间的角度，如图 8 所示。因为用于加工刀具和双圆弧中心的用于曲线拟合的中心位于通过接触点的法线方向线，内部的正常角度必须被转移为加工数据，以保证凸轮的精确形状。 nts用方程（ 14）可以更容易的定义如图 8 所示的接触点的位置角（c）。此外，滚子从动件和凸轮机构接触点的法线角（f）在图 8（ a）和同样在图 8（ c）中 ，由方程（ 15）得。法线的角度分别由式子（ 16）定义为往复运动平面从动件（图 8b）和方程（ 17）定义为振荡平底从动件（图 8d） 最后，内部正常的角度在凸轮接触点上可以表示成方程（ 18）为盘形凸轮机构如图 8. （ a） 往复式滚子从动件 nts （ b）往复式平面从动件 （ c）摆动滚子从动件 nts （ d）摆动平面从动件 图 8.盘形凸轮机构内部的正常角度 4.双圆弧插补 4.1 双圆弧曲线的特性 双圆弧插补是内部由 2 点，其中的双圆弧必须有一个在每个点处的切向分量的跨度连接来连接这两个圆弧。因此，所有的点上的双圆弧插补 曲线具有独特的斜率和朝向：双圆弧中心（这就是在本文中所谓的径向方向的角度，也有独特的方向角）。如图 9所示的径向方向的角度（ 21 和 ）可以定义为一个中心和一个连接的直线上的一个跨度之间的径向线的角度。双圆弧插补是深深依赖于径向方向角视图的准确性。 作者： Joong-Ho Shin, Soon-Man Kwon and Hyoungchul Nam 国籍： 韩国 出处： International Journal of Precision Engineering and Manufacturing ntsINTERNATIONAL JOURNAL OF PRECISION ENGINEERING AND MANUFACTURING Vol. 11, No. 3, pp. 419-427 JUNE 2010 / 419DOI: 10.1007/s12541-010-0048-6 1. Introduction Plate cam mechanism is a widely used machine component with the continuous contact motion of cam and follower, and can easily produce any functional motion of follower due to the rotation of cam. Cam mechanism has the diverse types by the combination of different shape of cam and motion of follower; plate or cylindrical cam, roller or flat-faced follower, and reciprocating or oscillating motion. In spite of the advantages of a few number of links, simple structure, positive motion, and compact size, cam mechanisms require the accurate shape design and precise machining procedures for satisfying the mechanical requirements. Under the low leveled design and manufacturing, cam mechanisms give the heavy effects on vibration, noise, separation, and overloading to an overall system. To avoid these effects, cam mechanism must be well designed accurately and machined precisely. Actually, a hybrid CAD/CAM approach may be the best solution that the shape data from the design process are directly combined to the machining data for the manufacturing process. Line interpolation and circular interpolation are commonly used in construction of the machining data from the profile data of cam. Line interpolation has the low accuracy and circular interpolation can not keep the accuracy because of the disconnective radii of curvatures or the discontinuous slopes at the connected point by two circular arcs as presented in Shin et al.1-3Recently, parameteric interpolation using B-spline and NURBS curve are suggested in Jung et al.5and Yang et al.6Also biarc interpolation is widely used and deeply dependent on the direction angle toward centers of biarc curves. Bolton7described a biarc curve based on the tangential angles at two points, Parkinson and Moreton8made a biarc curve based on a quadratic equation at three points, Meek and Walton9used spline types for constuction of biarc curve. Schonherr10introduced an approach to minimize the radii of biarc curves. Commonly these interpolation methods make the machining points increasing and then the excessive data for machining a curved shape make the machining errors increased. Thus, the precise machining process requires minimization of the machining points to keep the accuracy under a given machining tolerance. This paper introduces 3 steps of a hybrid CAD/CAM A Hybrid Approach for Cam Shape Design and Profile Machining of General Plate Cam Mechanisms Joong-Ho Shin1, Soon-Man Kwon1and Hyoungchul Nam1,#1 Department of Mechanical Design & Manufacturing, Changwon National University, #9, Sarim-dong, Changwon, Kyungnam, South Korea, 641-773# Corresponding Author / E-mail: nhchulchangwon.ac.kr, TEL: +82-55-267-1106, FAX: +82-55-267-1106KEYWORDS: Plate cam mechanism, Cam, Follower, Shape design, Profile machining, Contact point, NC data, Instant velocity center, Biarc curve fittingPlate cam mechanism can easily produce the positive and functional motions in contact of cam and follower. Generally cam mechanism is used in many fields of mechanical control, automation, and industrial machinery. To obtain the accurate motion of follower, the profile of cam must be designed and machined precisely. This paper proposes an instant velocity center method for the profile design and a biarc fitting method for the profile machining to 4 different types of plate cam mechanisms with reciprocating or oscillating motion and roller or flat-faced followers. The key of this paper is the introduction of a hybrid system combined the design procedures and the manufacturing procedures. The main idea is that the minimum machining data are built by the accurate biarc curves fitted directly from the design parameters. The radial direction angles toward biarc centers for the accurate biarc curve fitting can be defined directly by the contact angle of cam and follower given in the design procedures. An application of the proposed approach is verified the accurate profiles of a designed cam and a machining cam using the minimum NC data within a given machining tolerance. Manuscript received: July 16, 2009 / Accepted: February 18, 2010 KSPE and Springer 2010 nts420 / JUNE 2010 INTERNATIONAL JOURNAL OF PRECISION ENGINEERING AND MANUFACTURING Vol. 11, No. 3approach11for 4 different types of plate cam mechanisms. Firstly, the shape of cam is determined by the kinematic constraints at instant velocity centers and the contact angle at the contact point between cam and follower. The second step is to transform the contact angles into the center direction angles and then to calculate the radii of biarc curve. Finally, the machining data are minimized through expanding or contracting the biarc section whether the cam profile points are located inside or outside the range of a given machining tolerance. 2. Clarification of Accuracy on Biarc Fitting1-3Only the profile data of cam shape are defined in common design process of a plate cam. Then, the machining data must be developed by any curve fitting for NC (Numerical control) process. The circular fitting, which is most widely used in machining, has unreliability as shown in Fig. 1. A circle developed by three points 1(,P2,P3)P has a radius 1R and the other by points 2(,P3,P4)P has 2.R In the views of circular fitting two circles pass the profile points 1(,P2,P3,P4),P but the discontinuous slopes are made at these points and also the disconnective radii at points in mid-span. These defects make the fitted curve in low accuracy and then higher vibration in high speed operation of cam mechanism. P4P3P2P1O2O1R1R2S1S2SSlope 1Slope 2Fig. 1 Defects on circular fitting 3231R4R2P4P3O4O3S3S2R3S1P2O2O1P1R121112212Fig. 2 Continuous fitting by biarc Fig. 2 shows a continuous curve fitted by biarcs, which passes the profile points 1(,P2,P3,P4).P The biarc curve has 4 radii in this case. Radius 1R passes 1P to 1,S 2R for 1S to 2.S 3R for 2S to 3,S and 4R for 3S to4.P The slopes of the biarc curve are continuous and unique at every point. Also mid-points 1(,S2,S3)S are continuous without jump in radii. Thus, the biarc curve can keep the higher level of accuracy. As shown in Fig. 2, biarc fitting is highly dependent on radial direction angles (). The common design process of cam mechanism defines only the profile data and then machining process must use the angles from the circular fitting. This process gives the lower accuracy because of the incorrect angles. But the proposed approach in this paper can define the correct angles, which are given directly by design process of cam profile, and then keep the higher accuracy for the machining data. 3. Shape Design of Plate Cam 3.1 Displacement characteristics of cam mechanism For a plate cam mechanism with reciprocating roller follower shown in Fig. 3, the kinematic properties of follower motion can be defined as linear displacement ,Y first derivative ,Y and second derivative Y to the rotational angle c of cam. And the properties are given as angular displacement in case of oscillating follower. The instant velocity center method given in this paper uses the displacements and the 1st derivatives for determining the cam shape. CamFollowerContact point coordinateCam shape coordinatecCRSyxFig. 3 Plate cam mechanism with reciprocating roller follower 3.2 Shape design based on instant velocity centers4As shown in Fig. 4, Point Q is defined by a line through contact point C from roller center and a horizontal line and then it becomes instant velocity center. The velocity at point Q is proportional to a rotating speed of cam as in Eq.(1) and the velocity of roller at point R is defined in Eq.(2) as the linear velocity of follower. cQQdVLdt= (1) cRcdY dY dVdt d dt= (2) ntsINTERNATIONAL JOURNAL OF PRECISION ENGINEERING AND MANUFACTURING Vol. 11, No. 3 JUNE 2010 / 421By the kinematic characteristics of cam mechanism, the velocity at the instant velocity center ,QV is same as the velocity of follower .RV Thus, the velocity condition gives the location of the instant velocity center in Eq.(3). QcdYL Yd= (3) QR(Rx, Ry)LQVQVRcRrYxyC(Cx, Cy)Fig. 4 Contact position of cam and follower The contact angle shown in Fig. 4 is defined in Eq.(4) by a angle between a sliding velocity line and a normal line at a contact point of follower roller. The coordinates of the contact point are given in Eq.(5) where the coordinates of a roller center (,xR )yR can be calculated from the displacement ()Y and the geometric conditions (prime circle and eccentricity) of a given cam mechanism, and where rR is the radius of roller. Finally, the contact point (xC and )yC is given in Eq.(5) 1tanQxyL RR= (4) sinsinxxryyrCRRCRR=+=(5) QLQVQVfcYxC(Cx, Cy)F(Fx, Fy)Fig. 5 Plate cam with reciprocating flat-faced follower Fig. 5 shows a cam mechanism with reciprocating flat-faced follower. Instant velocity center Q is located on the horizontal line and defined in Eq.(6) based on the velocity conditions at instant velocity centers. Then, the contact point is defined in Eq.(7) QcdYLd= (6) x Qy yCLCF=(7) For a mechanism with oscillating roller follower as in Fig. 6, the distance of instant velocity center from cam center becomes in Eq.(8). The contact angle between cam and roller is expressed in Eq.(9) and then the contact point is defined in Eq.(10). Here, Zxyis the distance to a pivot from cam center. 1fxycQfcdZdLdd=+(8) 1tanQxyL RR= (9) sinsinxxryyrCRRCRR=+=(10) fQLQVQVRRZLZxycR(Rx, Ry)xyC(Cx, Cy)RrFig. 6 Plate cam with oscillating roller follower In a case of cam mechanism with oscillating flat-faced follower as shown in Fig. 7, the location of instant velocity center is formulated as in Eq.(11) and the contact point is given in Eq.(12) 1fxycQfcdZdLdd=+(11) ( )()2coscos sinx xy xy Q fy xy Q f fCZ Z LCZL= =(12) nts422 / JUNE 2010 INTERNATIONAL JOURNAL OF PRECISION ENGINEERING AND MANUFACTURING Vol. 11, No. 3fQLQVQVfVfZLZxycxyC(Cx, Cy)F(Fx, Fy)Fig. 7 Plate cam with oscillating flat-faced follower Finally, the profile of cam shape can be determined by transforming the contact point with the reverse angle of cam rotation as in Eq.(13), where xS and yS are the coordinates of cam profile. cos sinsin cosxx cy cy xcy cSC CSC C=+= +(13) 3.3 Internal normal angle at contact point The normal line at each contact point is shown in Figs. 4-7 for 4 different cases of the plate cam mechanisms. In this paper an internal normal angle () is defined as an angle between lines connected to cam center and to instant velocity center from contact point as shown in Fig. 8. Because tool centers for machining and biarc centers for curve fitting are located on the normal direction line through contact point, the internal normal angle must be transferred to the machining data process in order to guarantee the precise shape of cam. The position angle ()c of contact point shown in Fig. 8 is easily defined as in Eq.(14). Also the normal line angle ()f at contact point for cam mechanism with roller follower in Fig. 8(a) and Fig. 8(c) is same as in Eq.(15). The normal line angles are defined in Eq.(16) for reciprocating flat-faced follower (Fig. 8(b) and in Eq.(17) for oscillating flat-faced follower (Fig. 8(d), respectively. 1tanycxCC=(14) 1tanyyfxxR CR C=(15) 90fface slope angle = (16) 1tan 90yyfxxCZCZ=(17) Finally, the internal normal angle at contact point on cam profile can be expressed in Eq.(18) for plate cam mechanisms as shown in Fig. 8. f c = (18) CamFollowerfcRollercoordinateContact pointcoordinateCRyxairplane(a) Reciprocating roller follower CamFollowercfface slope angleContact pointcoordinateCyxQ(b) Reciprocating flat-faced follower CamFollowerfcRollercoordinateContact pointcoordinateCRyxQ(c) Oscillating roller follower CamFollowerfcContact pointcoordinatePivotcoordinateCZx, ZyyxQ(d) Oscillating flat-faced follower Fig. 8 Internal normal angles of plate cam mechanism ntsINTERNATIONAL JOURNAL OF PRECISION ENGINEERING AND MANUFACTURING Vol. 11, No. 3 JUNE 2010 / 4234. Biarc Interpolation74.1 Characteristics of biarc curve Biarc interpolation is to connect the two circular arcs inside of a span connected by 2 points, where the biarc must have one tangential component at each point. Thus, all of points on a biarc interpolated curve have the unique slope and also have the unique direction angle toward the biarc center (It is called by radial direction angle in this paper). The radial direction angles 1( and 2) in Fig. 9 can be defined angles between a radial line to a center and a connected line on a span. Biarc interpolation is deeply dependent on the radial direction angle in a view of accuracy. Biarc curve can be categorized into 4 different types by combination of radial direction angles in a span as shown in Fig. 9, i.e. Fig. 9(a) is case 1 1(0 Fig. 9(b) is case 2 1(0 and 20) and 20). Here, centers of a biarc curve in case 1 and case 2 are located in the same plane and biarc curve becomes continuous smoothly. Centers in case 3 and case 4 are positioned in the cross plane and a inflection point must be existed on biarc curve as shown in Fig. 9. 4.2 Definition of equation for biarc curve Fig. 10 shows biarc curves with same planar centers, where a radius 1R consists of an arc from point 1 to *S at a center 1O and a radius 2R makes an arc from *S to point 2 at a center 2.O The point *S is located on the common radial line. By connected two circular arcs continuously, all points on biarc curve have the continuous tangential components on the span with point 1 and 2. Biarc curve with the same planar centers in Fig. 10 have a convex or concave curve. By rearrangement of the radial direction angles 1( and 2), radii of arcs 1(R and 2)R and length of span (),L the equation for biarc curve with the same planar centers can be defined as in Eq.(19). ( )()12 1 2 1 2211222coscossinsin12cos cos 0RRLR R L +=(19) 1 2O1O2S*R1R2211 2O1O2R1R2S*21(a) Case 1 (b) Case 2 Fig. 10 Biarc with same planar centers 12O1O2R1R2S*2112O1O2R1R2S*21(a) Case 3 (b) Case 4 Fig. 11 Biarc with cross planar centers Fig. 11 shows biarc curves with cross planar centers, where a center 1O makes a circular arc with radius 1R in one plane and the other center 2O builds a circular arc with radius 2R in the opposite plane. Thus, a inflection point *S must be satisfied the conditions 12t1 t2n1 n21 2(a) Case 1 12t1t2n1 n21 2(b) Case 2 12t1t2n1n212(c) Case 3 12t1t2n1n212(d) Case 4 Fig. 9 Cases of biarc curves nts424 / JUNE 2010 INTERNATIONAL JOURNAL OF PRECISION ENGINEERING AND MANUFACTURING Vol. 11, No. 3of 3 points 1(,O2O and *)S located on the same line and the continuity of the tangential components of 2 circular arcs at *.S The equation for biarc curve with cross planar centers is defined in Eq.(20). ( )()12 1 2 1 2211222coscossinsin12cos cos 0RRLR R L +=(20) 5. Profile Machining Based on Biarc Curve 5.1 Radial direction angle for biarc curve On the profile of plate cam contoured by the contact points, any two consecutive points build a biarc span and can be connected by a biarc curve as shown in Fig. 12. Effectiveness of biarc interpolation depends on the accuracy of radial direction angle () at each point, because the centers of biarc curve are located on the lines defined by the radial direction angle at span points. 1XYP2P1n1211n222Fig. 12 Position angles, slope angle and length Arbitrary span on cam profile can be positioned at two points (1 and 2) by rotation of cam as shown in Fig. 12. Here, is the slope angle of a connected line between point 1 and point 2 and is the internal normal angle at each point defined in section 2.3. Position angles at span points 1(p and 2)p can be easily defined by the coordinates of the points from cam center. Thus, the radial direction angles at span points on biarc span are arranged as in Eq.(21). 11 122 2180pp =+ =+ (21) 5.2 Radius of curvature for biarc curve In the case of biarc curve with same planar centers shown in Fig. 13, the biarc equation of Eq.(19) can be rearranged as the following. 11212cos cos sin sin 1Z =+ (22) ( )2121 1 1 2 222coscos 0RRZ L R R L+= (23) The optimization of biarc curve requires the minimum difference of radii 1(R and 2).R The minimum difference is defined as in Eq.(24) and the radius of 2R is reformed as in Eq.(25). The differentiation of to 1R 1(/ 0)ddR= from Eq.(24) and Eq.(25) gives a quadratic equation as in Eq.(26) and also the radii of arcs 1(R and 2)R can be calculated in Eq.(27) and Eq.(25). ()212R R= (24) ()211211 22cososLRLRZR L=(25) ()2211 1 21221 2124cos2cos cos cos 0ZR LZ RLZ + + =(26) ()2 12111cos 1 cos2LRZ + = (27) In the case of biarc curve with cross planar center as shown in Fig. 14, the biarc equation of Eq.(20) is arranged as in Eq.(28) and Eq.(29). 2 12 12cos cos sin sin 1Z =+ (28) ( )2122 1 1 2 222coscos 0RRZ L R R L+= (29) 12RC1(XC1, YC1)R1R2RC2(XC2, YC2)L12S*Fig. 14 Radii on cross planar centers 12R1R212RC1RC2L(XC2, YC2)(XC1, YC1)S*Fig. 13 Radii on same planar centers ntsINTERNATIONAL JOURNAL OF PRECISION ENGINEERING AND MANUFACTURING Vol. 11, No. 3 JUNE 2010 / 425Here, it is assumed that the radii of arcs are proportional to the radial direction angles on biarc span as given in Eq.(31). By solvi
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