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In this paper, we attemptto manufacture the spiral bevel gear (SBG: the most complex type among the gear products) by a three-axis CNC milling machine interfaced with a rotary table. This consists of (a) geometric modeling of thespiral bevel gears, (b) process planning for NC machining, (c) a tool path planning and execution algorithmfor both 4-axis and 3/4-axis (three out of four axes) controls. Experimental cuts were made to ascertainthe validity and effectiveness of the presented method with a CNC milling machine controlled by the 3/4-axis control mode. 2001 Published by Elsevier Science Ltd.Keywords: Gear manufacturing; Spiral bevel gear; Geometric modeling of gears; Sculptured surface machining; Rotarytable application; Additional-axis machining technology1. IntroductionGears are efficient and precision mechanisms for industrial machinery as a means for powertransmission. Among the various types of gears (Fig. 1), the spiral bevel gears (SBG) are themost complex type and are used to transmit the rotational motion between angularly crossedshafts. Previous studies on gears have been mainly concerned with the design and analysis ofgears. The geometric characteristics and design parameters of SBGs have been studied extensively* Corresponding author. Fax: +82-54-279-5998.E-mail address: shspostech.ac.kr (S.H. Suh).0890-6955/01/$ - see front matter 2001 Published by Elsevier Science Ltd.PII: S0890-6955(00)00104-8834S.H. Suh et al. / International Journal of Machine Tools & Manufacture 41 (2001) 833850Fig. 1.Various industrial gears.in 19. In 6, Tsai and Chin presented a mathematical surface model of the bevel gears (straightand spiral bevel gears) based on basic gearing kinematics and involute geometry along withtangent planes. Later, this model was compared with another model, based on exact sphericalinvolute curves, by Al-Daccak et al. 7. Recently, Shunmugam et al. 8,9 presented a differentmodel, and its accuracy (compared with the SBG manufactured by the special machine tools)was verified in terms of the normal deviation.As far as manufacturing is concerned, almost all previous works have assumed that the gearsare machined by a special type of machine tools, such as gear hobbing and shaping machines.This may be why literature on gear manufacturing is sparse in the open research domain. In fact,no research results on tool path planning for sculptured surface machining (SSM) of SBGs canbe found, although some authors have remarked on the possibility of gear manufacturing by theCNC milling machine 6. Recently, CNC based gear manufacturing machine tools have beendeveloped and increasingly used in industrial practice (Fig. 2). However, their kinematic structureis still inherently different from the industrial CNC milling machine, as the former is designedfor a special type of cutter.In this paper, we attempt to manufacture SBGs by the SSM method using a three-axis millingmachine interfaced with a rotary table. In terms of production rate, it is obvious that the SSMmethod will be lower than that of the special machine tool. Other than production rate, the SSMmethod is advantageous in the following respects: (1) the conventional method requires a largeinvestment for obtaining various kinds of special machinery and cutters dedicated to a very limitedclass of gears in terms of sear type, size, and geometry; (2) by the SSM method, a broad rangeFig. 2.Special machine tools and cutters for manufacturing SBGs.835S.H. Suh et al. / International Journal of Machine Tools & Manufacture 41 (2001) 833850of gears can be manufactured with the industrial CNC milling machine; (3) a special type of gear,for instance “huge” gears of diameter over 1000 mm, and “crown” gears can be machined bythe SSM method, which cannot be done by the dedicated gear machine tools, except in verylimited cases.In view of the above, special attention is given to the capability of the SSM method in termsof geometric accuracy and surface quality together with machining time. If the performance iscomparable, except for the production rate, the SSM method can be applied in industrial practicefor NC machining of huge SBGs, while the production rate is not emphasized. This paper presentscomprehensive technology including geometric modeling, process planning, tool path algorithms,and experimental validation.2. Geometric modeling of the SBGsTypically, geometric specification of SBGs is given by a set of parameters. These parametersare provided with an engineering drawing, as shown in Fig. 3. Some parameters (principalparameters) are required for defining the geometry, while others (auxiliary parameters) can bederived based on a formula. Table 1 summarizes some of the crucial parameters including relation-ships among parameters 2.Using the parameters, the surface model can be derived as follows. As illustrated in Fig. 4, thesurface between two teeth is modeled by the large section curve swept along the spiral curve.The section curve is decomposed into five fragments; Si(ui, i P 1:5, where uiP 0, umaxi) is theparameter for fragment i. Denoting w by the parameter along the spiral curve, the surface modelcan be represented by Si(u, w), i P 1:5 as shown in Fig. 4.Siand S5are the involute surfaces, and S2and S4the filleted surfaces, and S3the bottom surface.Fig. 3.An engineering drawing.836S.H. Suh et al. / International Journal of Machine Tools & Manufacture 41 (2001) 833850Table 1Principal and auxiliary geometric parameters of Gleason SBGs at large sectionPinionGearPrinciple parametersModuleMNumber of teethZ1Z2Face widthbPressure angleanShaft angleSSpiral anglebHand of spiralH (left or right)BacklashBTeeth thickness constantKAuxiliary parametersEfficienthe=1.700mWhole teeth heighth=1.888mPitch circle diameter (PCD)d1=z1md2=z2mPitch cone angled1=tan1(z1/z2)d2=902d1Cone distanceR=d2/(2 sin d2)Circular pitcht0=pmAddendumhk1=he2hk2hk2=0.460 m+0.390 m/(z2/z1)Dedendumhf1=h2hk1hf2=h2hk2Pitch to dedendumqf1=tan1(hf1/R)qf2=tan1(hf2/R)Addendum cone angledk1=d1+qf2dk2=d2+qf1Dedendum cone angledf1=d12qf1df2=d22qf2Outer diameterdk1=d1+2hk1cos d1dk2=d2+2hk2cos d2Distance from apexY1=d2/22hk1sin d1Y2=d1/22hk2sin d2Arc thicknesss1=t02s2S2=t0/22(hk12hk2)(tan an/cos b)2KmAxial distance along addendumX1=b cos dk1/cos qf2X2=b cos dk2/cos qf1Inner diameterdf1=dk122b sin dk1/cos qf2df2=dk222b sin dk2/cos qf1Blank angle (1)qx1=902qf2qx2=902qf1Blank angle (2)qy1=902d1qy2=902d2Base circle radiusr1=d1cos(an)r2=d2cos(an)Base cone angledb1=sin1(r1/2R)db2=sin1(r2/2R)837S.H. Suh et al. / International Journal of Machine Tools & Manufacture 41 (2001) 833850Fig. 4.Parametric representation of tooth surface.S2, S3, and S4provide clearance during the motion, and hence are of relatively little geometricalimportance. The involute surfaces are the crucial surfaces where the gear and pinion (note: gearsare operated pairwise, and the larger is called the gear, and the smaller is the pinion) are continu-ously contacted during the rotational motion. In what follows, we present a method for derivinga surface model for S5of the pinion (note that S1for the pinion, and S1and S5for the gear canbe similarly derived).The involute curve used for the SBG is defined on the sphere, namely the spherical involutecurve. Consider the large section of the spiral bevel pinion, where w is the cone distance R inTable 1. Then, a reference circle of radius w and a base circle radius of r1=d1cos(an), where d1and anare respectively the inner diameter of the pinion and the pressure angle (Table 1), can bedefined as shown in Fig. 5. Parameterizing the points on the base circle by angular distance fromthe reference point, the points can be uniquely defined by parameter u. Further, for the pointparameter of u, the contact circle radius of w centered at the origin of the reference circle canbe uniquely defined. Then, the spherical involute point (P) at u is on the contact circle, the arclength of which is the same as that along the base circle (see Fig. 5). By changing u from 0 toumax(defined below), and w from R (large section) to R2b (small section, where b is the facewidth), the spherical involute curve for the straight bevel pinion can be obtained,umax5cos1Stan1(dk1/2X1)cos(db1)Dsin(db1)(1)where db1is the base cone angle.To represent P with respect to reference coordinate frame A, the position of P with respect tothe B coordinate frame is transformed as follows:P5RABPB(2)where838S.H. Suh et al. / International Journal of Machine Tools & Manufacture 41 (2001) 833850Fig. 5.Spherical involute curve for the given w.RABP R335RotX, uRotY, db1.(3)andPB5w cos(u sin(db1), 2w sin(u sin(db1), 0T(4)In the spiral bevel pinion, the spherical involute curve is rotated along the spiral curve. Amongthe widely used spiral curves, logarithmic, circular, involute, the circular cut spiral curve is usedin this paper (as in Gleason 2). As illustrated in Fig. 6, two coordinate frames are defined: OFig. 6.Spiral curve.839S.H. Suh et al. / International Journal of Machine Tools & Manufacture 41 (2001) 833850and C(H, V), where O is defined on the pitch plane and C(H, V) set on the center of the circularcutter. Based on 6, q0 determined by w can be derived as follows:q052 tan13VH2+V2U2H+U4(5)whereU5w2+R2m2RmRcsin(b)2w.(6)q0 of Eq. (5) is represented on the pitch plane coordinate frame, and it can be transformed intoq in the A coordinate frame (Fig. 5) as follows:RA5RotX, q531 000 cos(q) sin(q)0 sin(q) cos(q)4(7)whereq5q0sin(d1)(8)Note that the parameter for the spiral curve (w) determines not only the size of the sphericalinvolute curve but also the rotational amount of the spiral curve. Based on the discussion so far,the bi-parametric surface model for the spiral pinion can be derived as follows.S1(u, w)5RAPA5RARABPB5w(9)3C(db1)C(uS(db1)C(q)S(u)S(db1)C(uS(db1)C(u)S(uS(db1)S(q)C(u)S(db1)C(uS(db1)+S(u)S(S(db1)S(q)S(u)S(db1)C(uS(db1)C(u)S(uS(db1)+C(q)C(u)S(db1)C(uS(db1)+S(u)S(uS(db1)4where u P 0, umax.840S.H. Suh et al. / International Journal of Machine Tools & Manufacture 41 (2001) 8338503. Manufacturing the SBGsTo manufacture the SBGs with the CNC milling machine, tool path planning is the key toobtaining successful results. In tool path planning, various aspects should be taken into consider-ation: (1) the geometric accuracy and surface quality of the machined surface, (2) the time formachining, and (3) the configuration of the machine tool for manufacturing.3.1. Machine tool configurationAs far as machine tool configuration is concerned, it is obvious that a rotational motion of thetool is required for NC machining of the SBGs. Based on the machinability analysis 10, at leastfour-axis controls are required for NC machining of SBGs by one set-up. Thus, a rotary table tobe interfaced with the three-axis milling machine is required. Depending on the capability of themachine tool controller: (a) all the four axes (one-axis motion for the rotary table and three-axismotion for the cutting tool) can be simultaneously controlled; (b) only three axes out of the fouraxes can be controlled simultaneously. The latter is called the additional-axis machine system,which can often be found in industrial practice where the rotary table is controlled by the fourthaxis of the three-axis machine tool controller (see 11 for details). In this paper, we present atool path algorithm for both configurations.3.2. Machining strategyThe workpiece is premachined as a conic form by turning operation. The volume to be removedisthesweptvolumeofthecross-sectionCUIWalongthespiralcurvedefinedbySi(u, w), i P 1:5). The volume is removed by several processes: (1) rough cut with several flatendmills; (2) semi-finished cut with several ball endmills; and (3) finish cut with a ball endmill.To minimize the machining time, a larger tool is desired for the rough cut and semi-finish cut.The finish cut allowance is set (for instance 0.3 mm), and the semi-finish cut removes the unevensurface (resulting from the rough cut). During the finish cut, the whole surface is machined by asingle ball endmill of diameter D=0.8I (this is based on a heuristic), where I is the chodal lengthbetween the two points defining the S3curve in the small section curve (see Fig. 4). This is toprevent any cutter marks on the surface due to tool change. Tool path algorithms for rough andsemi-finished cuts are omitted for the brevity of the paper.3.3. Tool path planning for finish cutThe surface model S1(u, w) is machined by a ball endmill of radius R. As mentioned earlier,the involute surfaces S1and S5are the most important surfaces, the accuracy of which should bestrictly controlled. Our method is based on the CC-parametric scheme, where the CC-points aresampled from the parametric surface model, equally distanced on the parametric plane. For thesake of machining efficiency, tool motion along the w direction is chosen. In what follows, inter-ference-free CL-data for S5(u, w) is presented, as the same can be applied for S1(u, w).841S.H. Suh et al. / International Journal of Machine Tools & Manufacture 41 (2001) 8338503.3.1. Dealing with tool ball interference (TBI)For the CC-point on the iso-parametric curve of u, the tool center point is:C(u, w)5S(u, w)1RN(u, w)(10)where N(u, w) is the unit normal vector at S(u, w). Considering that: (a) the tool size is smallerthan the small side cross-section (D=0.8I), and (b) S1(u, w) is convex, interference due to the toolball (TBI) is not present for most CC-points, except those close to S2. Precisely speaking, theCC-point at the boundary region is the only place where TBI may occur (Fig. 7). If TBI isdetected, the CC-point should be removed for subsequent machining with a smaller tool size.Alternatively, one may allow tool gouge in this region, because (a) the overcut area is the S3region, (b) which is to provide clearance between the gear and the pinion, and hence (c) a slightovercut in this region is allowed (this will provide more clearance). Based on the above compu-tationally efficient interference detection and handling, the algorithm can be developed.3.3.2. Dealing with tool-axis interferenceIn addition to TBI, there is another type of tool interference called TAI (tool interference dueto the tool axis) in multi-axis machining, where the tool orientation is changed. TAI can beavoided by changing the tool orientation so that the tool body does not interfere with the surfacegeometry. TAI can be handled in two ways: (a) detection of tool body interference, followed byadjustment of the tool-axis; and (b) finding a feasible range of the interference-free tool axis,followed by selecting an interference-free axis. We take the second approach. In what follows,we present a computationally efficient method for finding a feasible range defined by the twobounding axes A1and Arfor the given CC-point S1(u, w) as shown in Fig. 8.Suppose the tool center point and its unit normal vector are given by C and NC. Then, the toolmotion in the four-axis configuration is defined on the CL-plane:PC=PuPx=Cx, where Cxis thex value of C. Let C1i, i P 1:m, C2j, j P 1:n be the offset points on the CL-plane (Fig. 9). Notingthat T1P C1i, T2P C2i, consider the problem of finding Ti. Define the reference axis (Fig.9(a) asV5CC113NC(11)Fig. 7.TBI in the boundary region.842S.H. Suh et al. / International Journal of Machine Tools & Manufacture 41 (2001) 833850Fig. 8.Two bounding axes.Fig. 9.Determination of the two bounding axes.and the left (respectively, right) tangent line at C, Al, (resp. Ar), as the rotation of NCabout thereference axis as follows:Al2NCRotFV, 2p2G(12)Ar5NCRotFV,p2G(13)Initializing T1with a point on the left-tangent line, T1is updated by C1i, if the following updatecondition is satisfied:(CT13C1i)V.0(14)This condition is derived from the following geometric insight: defining qias the angle (CCW is843S.H. Suh et al. / International Journal of Machine Tools & Manufacture 41 (2001) 833850positive with respect to the reference axis) to rotate Alto align with CC1i(see Fig. 9(b), the firstcritical point (T1) is the offset point with the largest angle. Thus, the current T1(of C1a) is updatedby C1b(Fig. 9(b), but T1is not updated by C1c. The search for T1proceeds from C11throughC1m. The procedure for finding T1is very similar except for the update condition:(CT23C2j)V,0, j P k11:n(15)where C=C2k3.4. NC code generation for simultaneous four-axis controlFor the CC-points S5(u, w), what we have obtained up to this point are: gauge-free tool centerpoint C, and the feasible tool axis range in terms of two critical points T1and T2. From these,the NC code ttx, tty, ttz, a, where the first three components are the tool tip position, and a isthe rotation angle of the rotary table, can be computed as follows.Noting that the feasible range is the range through which the tool axis can vary without inter-ference, it can be thought of as a cone defined by C and two critical axes, Aland Ar, passingthrough the tool center point and two critical points, T1and T2(Fig. 8). With the two criticalpoints, an interference-free tool axis can be readily determined as the medial axis (unit vector)of the cone;A5uCT2uCT1+uCT1uCT2uCT2uCT1+uCT1uCT2(16)In general, the tool axis vector is not aligned with the spindle axis (Z). In the four-axis con-figuration where the workpiece is oriented by the rotary table, it is necessary to align the toolaxis vector with the spindle axis. The rotation angle to access a CC-point S5(u, w) is determinedsuch that the tool axis vector A(u, w) is parallel to the XZ-plane. Decomposing the tool axis vectorinto Ax, Ay, Azthe rotation angle (Fig. 10) isFig. 10.Tool-axis determination for four-axis control.844S.H. Suh et al. / International Journal of Machine Tools & Manufacture 41 (2001) 833850a5 tan1SAyAzD2p2(17)Rotating the workpiece by a, the tool tip position tt(u, w) is as follows:tt(u, w)5S5(u, w)1RA(u, w) RotX, a2RK(18)where K=0, 0, 1T, and RotX, a is a 33 matrix indicating rotation of a with respect to theX-axis.Solving Eq. (18) for ttx, tty, and ttz:ttx5Cx(19)tty5cos(a)Cy1 sin(a)Cz(20)ttz52sin(a)Cy1 cos(a)Cz2R(21)3.5. NC-data generation for 3/4-axis controlTo execute the NC code given in the above, simultaneous four-axis control is required, as allthe four coordinates of the NC code may change in every block. Thus, the above NC code cannotbe executed by the 3/4-axis control, where at most three axes (out of four axes) can be simul-taneously controlled. As described earlier, 3/4-axis control is often present in industrial practicewhere the rotary table is interfaced with the fourth axis port of the machine tool controller.For 3/4-axis control, one of the four coordinates must be kept constant. In this research, wekeep the y coordinate of the tool tip position constant as y*. For four-axis control, we take themedial axis of the feasible region as the tool-axis. For 3/4-axis control, the medial axis cannotbe taken as the tool axis any more, as the y value of the tool tip position (in the machine coordinateframe) would not be the same for all the blocks. Our method to determine the tool axis vectoryielding the same y value is as follows (Fig. 11)Step 1: For CCi, i P 1:N, where CCiis the ith CC-point along the iso