WIXLIB

multi-structure features whether or not. So planar edge curve is adopted, i.e., thecircular arc edge is the intersecting planar curve of the circular torus and the planecomposed of an extended line of straight edge and a tangential line at the end of theconical helix edge.As shown in Fig. 2, the revolving surfaces of the cutting edges on TEMC aredefined as three parts: a circular truncated conical surface A where the conical helixedge a lies, a circular torus B where the circular arc edge b lies, and a concave conicalsurface C. Their relationship is that B is tangent with A and C. Straight edge c has themulti-structure features of tooth offset center and introversion.Define I as the intersection of a and b, J as the intersection of b and c. Define L1as the extended line of c, L2 as the tangential line of a at I, L3 as the intersecting lineof B and C, L4 as the intersecting line of A and B. Define P as the intersection of L1and L2. Plane M is composed of L1 and L2. The bottom edge consists of b and c.Fig. 2Relevant geometrical elements of bottom edge2.1 Parametric model of edge cTo reduce the complexity of mathematical modeling, define the coordinatesystem [O1-X1Y1Z1] as the first coordinate system, and the origin O1 as center of B. Z1is the cutter axis and the positive direction is from the cutter shank to the tip. X1 isparallel to the projection of c on the plane X1Y1. The point coordinates in the firstcoordinate system are identified by subscript 1.

Let γ be the angle between X1-axis and the projection of IO1 on the plane X1Y1.Rotate the first coordinate system on Z1-axis by angle γ negatively, and the coordinatesystem [O2-X2Y2Z2] is obtained as the second coordinate system. The pointcoordinates in the second coordinate system are identified by subscript 2. Then, theparametric models of c and b can be described in the second coordinate system.As shown in Fig. 3, define L as the intersection of L3 and the plane X1Z1, Q as thetip of C, K as the endpoint of c, Or as the center of the cross section of B on the planeX1Z1. Let re be the section radius of B, R be the distance between I and the cutter axis,κ be the half cone angle of A, β be the helix angle of a at I (angle between the helixedge on the revolving surface and the generator curve).Fig. 3Schematic diagram for the design of edge cOn the basis of cutter geometry relationship, the coordinates of J and K in thefirst coordinate system are as follows x J 1 J 1 y J 1 z J 1 xK 1 K 1 y K 1 z K 1 (x r cosη eOr 12 re sin η ) h 2 ,h re cosη . 22( xOr 1 re sinη ) h tanη 0h(1)(2)According to the direction of the coordinate system, the rotation transformationmatrix from the first coordinate system to the second coordinate system can be

expressed as follows cos γT1 2 sin γ 0sin γcos γ00 0 . 1 (3)Obviously, the coordinates of I can be described as xI 2 I 2 y I 2 z I 2 R 0 , re sin κ (4)where: R xOr 1 re cos κ .Let τL1 2, τL2 2 be the unit direction vector of L1 and L2 respectively. Then τL1 2and the parametric equation L1 2(t1) of L1 can be obtained by Eqs. (5) and (6)respectivelyτ L1 2 τ L1 x 2 cos γ cosη sin γ cosη , τ L1 y 2 τ L1z 2 sin η x xJ 2 t1 τ L1x 2 , L1 2 ( t1 ) y y J 2 t1 τ L1 y 2 , z z t τJ 2L1 z 2 .1 (5)(6)where t1 is the independent variable.Similarly, τL2 2 and the parametric equation L2 2(t2) of L2 can be obtained by Eqs.(7) and (8) respectively τ L 2 x 2 τ L 2 2 τ L2 y 2 τ L 2 z 2 cos β sin κ ,sin β cos β cos κ x xI 2 t2 τ L 2 x 2 , L2 2 ( t2 ) y yI 2 t2 τ L 2 y 2 , z z t τI 22L2z 2. where t2 is the independent variable.For L1 and L2 intersect at P, then(7)(8)

xP xJ 2 t1 τ L1x 2 xI 2 t2 τ L 2 x 22 yJ 2 t1 τ L1 y 2 yI 2 t2 τ L 2 y 2 . yP 2 z J 2 t1 τ L1z 2 z I 2 t2 τ L 2 z 22 zP (9)So t1, t2 and γ can be obtained by Eqs. (5) to (9), shown as follows ( yI 2 t2 τ L 2 y 2 )( xJ 1 t1 cosη ) ( xI 2 t2 τ L 2 x 2 ) yJ 1 , sin γ 2yJ2 1 ( xJ 1 t1 cosη ) ( xI 2 t2 τ L 2 x 2 )( xJ 1 t1 cosη ) ( yI 2 t2 τ L 2 y 2 ) yJ 1 , cos γ 2yJ2 1 ( xJ 1 t1 cosη ) where: t1 t2 T1 (10) T2 T22 4T1 T3；2T1z J 1 z I 2 t1 sin ητ L2z 2(τ2L2x 2； τ L22 y 2 ) sin 2 η τ L22 z 2 cos 2 η ； xJ 1 τ L22 z 2 cosη τ L 2 x 2 ( xI 2 τ L 2 z 2 z I 2 τ L 2 x 2 z J 1 τ L 2 x 2 ) ；T2 2 τ L 2 y 2 ( yI 2 τ L 2 z 2 z I 2 τ L 2 y 2 z J 1 τ L 2 y 2 ) sin η T3 (x τ L 2 z 2 z I 2 τ L 2 x 2 z J 1 τ L 2 x 2 ) 2I 2( yI 2 τ L 2 z 2 zI 2 τ L 2 y 2 zJ 1 τ L 2 y 2 ) ( xJ2 1 yJ2 1 )τ L22 z 22.For c and L1 are collinear, the unit direction vector τc 2 and parametric equationc 2(t1) of edge c can be obtained by Eqs. (5) and (6) respectively, and the range of t1 is[0, xJ 2 τ L1 x 2 ].2.2 Parametric model of edge bAs shown in Fig. 4, define E as an arbitrary point on B, E′ as the rotatingprojection of E on the plane X1Z1, N as an arbitrary point on bottom edge. Let θ be theangle between E′Or and X1-axis (counter clockwise direction is positive), φ be theangle between the projection of EO2 on the plane X2Y2 and X2-axis (clockwise

direction is positive).Fig. 4Schematic diagram for the design of edge bThe normal vector nM 2 of M is determined by the vector cross-product of τL1 2and τL2 2 as followsnM 2 nMx 2 sin β sin η cos β cos κ cosη sin γ nMy 2 τ L1 2 τ L 2 2 cos β cos κ cosη cos γ cos β sin κ sin η , nMz 2 cos β sin κ cosη sin γ sin β cosη cos γ (11)Hence, the equation of M can be obtained by Eq. (12)nMx 2 ( x xI 2 ) nMy 2 ( y yI 2 ) nMz 2 ( z z I 2 ) 0.(12)By the definition of the circular torus[26], the parametric equation of B can bewritten as x ( xOr 1 re cos θ ) cos φ , π B 2 (θ , φ ) y ( xOr 1 re cos θ ) sin φ , κ θ η , 0 φ 2π .2 θzrsin, e(13)Since b is defined as a planar curve, then b is the intersecting line of M and B andangle φ can be determined by Eqs. (12) and (13)22 22 sin φ nMy 2 (T4 nMz 2 re sin θ ) nMx 2 ( nMx 2 nMy 2 ) ( xOr 1 re cos θ ) (T4 nMz 2 re sin θ ) , ( nMx2 2 nMy2 2 ) ( xOr 1 re cos θ ) 2222 nMx 2 (T4 nMz 2 re sin θ ) nMy 2 ( nMx2 nMy 2 ) ( xOr 1 re cos θ ) ( T4 nMz 2 re sin θ ) cos φ , ( nMx2 2 nMy2 2 ) ( xOr 1 re cos θ ) (14)

where:T4 xI 2 nMx 2 yI 2 nMy 2 z I 2 nMz 2 .Substituting Eq. (14) into Eq. (13) gives the parametric equation b 2(θ) of edge b nMx 2 (T4 nMz 2 re sin θ ) nMy 2 x nMy 2 (T4 nMz 2 re sin θ ) nMx 2 ) b 2 (θ y z re sin θ , (n2Mx 2(n2 nMy2 ) ( xOr 1 re cos θ ) ( T4 nMz 2 re sin θ )22Mx 2(n2Mx 2(n22 nMy2 ) ( xOr 1 re cos θ ) ( T4 nMz 2 re sin θ )2Mx 2,2 nMy2 )2 n2My 2)2π , κ θ η .2 (15)Take the derivative of Eq. (15) with respect to θ, the unit direction vector τb 2(θ)of edge b can be expressed as 22 nMy 2 T6 T7 sin θ nMy 2 nMz 2 T5 cos θ nMx 2 nMz 2 T6 T7 T5 cos θ 2T6 T6 T7 2 T5 2 (T5 sin θ T7 nMz 2 cos θ ) τ bx 2 22π nMx 2 T6 T7 sin θ nMx 2 nMz 2 T5 cos θ nMy 2 nMz 2 T6 T7 T5 cos θ τ b τby 2 , κ θ η ,2 (θ ) 2222 T6 T6 T7 T5 (T5 sin θ T7 nMz 2 cos θ ) τ bz 2 T6 T7 2 T5 2 cos θ 222 T6 T7 T5 (T5 sin θ T7 nMz 2 cos θ ) (16)where: T5 T4 nMz 2 re sin θ ；22 T6 nMx2 nMy 2 ；T xOr 2 re cos θ .72.3 The generalized helix angle of edge bThe parametric model of edge b is already obtained. Although edge b is a planaredge curve, it can be also regarded as a curve with a variable helix angle. At present,the generalized helix angle on the revolving surface has two definitions[24]: one is theangle between the helix edge on the revolving surface and the generator curve[22], theother is the angle between the helix edge on the revolving surface and the cutteraxis[25]. The first definition is considered in this paper. The unit tangent vector of thegenerator curve of B at E can be expressed in the following form

sin θ cos φ τ BE 2 (θ ) sin θ sin φ cos θ (17)Hence the generalized helix angle at N can be given byβ r arccosarccosτ BE 2 (θ ) τ b 2 (θ )τ BE 2 (θ ) τ b 2 (θ )T6 T7 2 T5 2T6 T7 2 T5 2 (T5 sin θ T7 nMz 2 cos θ )2π , κ θ η .2 (18)2.4 Parametric model of edge aAs shown in Fig. 5, define F as an arbitrary point on a. Let ϕ be the anglebetween the projection of FO2 on the plane X2Y2 and X2-axis (clockwise direction ispositive).Fig. 5Schematic diagram for the design of edge aBy the definition of the generalized helix, the parametric equation a 2(ϕ) of edgea can be written as

x xI 2 e ϕ sin( κ ) cot β cos ϕ , ϕ sin( κ ) cot β sin ϕ ,a 2 (ϕ ) y xI 2 e x 1 e ϕ sin( κ ) cot β z I 2 re sin κ , tan( κ ) (ϕ 0) .(19)Take the derivative of Eq. (19) with respect to ϕ, the unit direction vector τa 2(ϕ)of edge a can be expressed as T8 x ,2 T8 T92 T102 τ ax 2 T9 τ a 2 (ϕ ) τ a y 2 ,y 222 TTT 8910 τ az 2 T10 z ,222 TTT8910 (ϕ 0) .(20)where: T8 sin κ cot β cos ϕ sin ϕ ；T9 sin κ cot β sin ϕ cos ϕ ；T10 cos κ cot β . 3 Grinding method of bottom edgeParametric model of the bottom edge is the base of the tool path of the grindingwheel. Bottom edge is the intersection curve of rake face and flank face, and itsstructural parameters are guaranteed by the grinding process of the two faces. ForTEMC has a complex structure and lots of parameters and the purpose of the grindingmethod in this paper is to guarantee the precision of parameters related to bottom edge,the detailed method of the section profile formation was not presented here forsimplification purpose. For convenience, the tool path calculation is deduced in thefirst coordinate system thus the origin of programming is O1 and grinding wheel wear

is not considered.3.1 Parameters of the grinding wheelThe profile of V-shaped grinding wheel and Cup-shaped grinding wheel can bothbe defined by three parameters. Let Rg (the radius of the big-end of the grindingwheel), Hg (the thickness of the grinding wheel) and αg (the taper angle of thegrinding wheel) be three given geometric parameters of the grinding wheel profile,shown in Fig. 6.Define the coordinate system [Og-XgYgZg] as the grinding wheel coordinatesystem, and the origin Og as the center of the big-end of the grinding wheel, Zg as thegrinding wheel axis. Let n 1 (the unit direction vector of Zg-axis) and P 1 (thecoordinates of the origin Og) be two parameters of tool path in the first coordinatesystem. Define ν as the inclined angle in grinding process. Let ν be ν′ in rake facegrinding and let ν be ν′′ in flank face grinding.Fig. 6Geometric parameters of the profile of grinding wheel3.2 Grinding tool path for rake faceThe grinding of the rake face of bottom edge is to form the chip-breaker-grooveof the end cutting edge and the normal rake angle of TEMC.Define the coordinate system [On-XnYnZn] as the movable normal-section localcoordinate system, the origin On coincides with N, Zn-axis is tangent to the bottomedge and Xn-axis and the cutter axis lies in the same plane. In the grinding process, ν′(the inclined angle restrained by the depth of chip-breaker-groove in rake facegrinding) and γn (the normal rake angle of bottom edge) are pre-designed.In a rake face grinding process, the coordinate system [O′g-X′gY′gZ′g] can be

obtained from the coordinate system [On-XnYnZn] by the following steps: rotate thecoordinate system [On-XnYnZn] around Zn-axis by angle γn positively, then aroundYn-axis by angle ν′ negatively, around Xn-axis by 90 negatively, at last translate therotated coordinate system from On to O′g to get the grinding wheel coordinate system.Let N 1 [xN 1, yN 1, zN 1]T be the coordinates of N in the first coordinate system.Define T1-n and R1-n as translation and rotation homogeneous transformationmatrix(HTM) from the coordinate system [O1-X1Y1Z1] to [On-XnYnZn] respectively,T′n-g and R′n-g as translation and rotation HTM from the coordinate system [On-XnYnZn]to [O′g-X′gY′gZ′g] in rake face grinding process respectively as shown in Fig. 7. Theycan be obtained by Eqs. (19) to (22)T1 nR1 n cos θ 0 sin θ 00 sin θ100 cos θ00 1 0 0 0R'n g sin γ ncos γ n000 00 10 0 0 cos β r 0 0 sin β r 1 00T'n g cos γ n sin γn 0 0xN 1 yN 1 z N 1 1 0 01 00 10010 1 0 0 00sin β rcos β r0(21)0 0 π ，κ θ η 0 2 1 0 0 0 1 0 Rg 0 1 0 0 0 1 0 cosν 10 00 sinν 1 1 00 sinν 1100 cosν 100(22)(23)0 1 00 0 00 0 1 1 0 001000 0 0 1 (24)

Fig. 7Schematic diagram for rake face grinding(1) The vector of the grinding wheel axisAccording to the HTM theory, the vector of the grinding wheel axis in rake facegrinding can be expressed as followsn' 1 R1 n R'n g [ 0 0 1 0]T(25)(2) The position of CL pointAccording to the HTM theory, the position of CL point in rake face grinding canbe expressed in the following formP' 1 T1 n R1 n R'n g T'n g [ 0 0 0 1]T(26)3.3 Grinding tool path for flank faceThe grinding of the flank face of bottom edge is to form the normal relief angleof TEMC.In the grinding process, ν′′ (the inclined angle to avoid interference betweengrinding wheel and other tooth in flank face grinding) and αn (the normal relief angleof bottom edge) are pre-designed.In a flank face grinding process, the coordinate system [O′′g-X′′gY′′gZ′′g] can beobtained from the coordinate system [On-XnYnZn] by the following steps: rotate thecoordinate system [On-XnYnZn] around Zn-axis by angle αn negatively, then around

Xn-axis by angle ν′′ positively, around Yn-axis by 90 negatively, at last translate therotated coordinate system from On to O′′g to get the grinding wheel coordinate systemfor flank face grinding.Define T′′n-g and R′′n-g as translation and rotation HTM from coordinate system[On-XnYnZn] to [O′′g-X′′gY′′gZ′′g] in flank face grinding process respectively as shownin Fig. 8. They can be obtained by Eqs. (25) and (26)T''n gR''n g cos α n sin αn 0 0sin α ncos α n001000 Rg 2 0 0 1 0 0 1 0 0 10 0 0 0 cosν 21 0 0 sinν 2 0 1 000Fig. 8 1 0 0 0(27)0 sinν 2cosν 200 00 00 1 1 00 1 0 1 0 0 0 0 0 0 0 1 (28)Schematic diagram for flank face grinding(1) The vector of the grinding wheel axisAccording to the HTM theory, the vector of the grinding wheel axis in flank facegrinding can be expressed as followsn'' 1 R1 n R''n g [ 0 0 1 0]T(2) The position of CL point(29)

According to the HTM theory, the position of CL point in flank face grinding canbe expressed in the following formP'' 1 T1 n R1 n R''n g T''n g [ 0 0 0 1]T(30)4 Experiments and results4.1 Modeling experimentTo verify the validity of the proposed parametric modeling method, modelingexperiment was carried out in MATLAB. To validate the G1 continuity at the twojoints of a circular arc edge with the connected straight edge and conical helix edge,one TEMC with multi-structure features of tooth offset center and introversion of itsstraight edge was taken for example. The design values related to parametricmodeling of the cutting edge are shown in Table 1.Table 1Design values related to parametric modeling of the cutting edgeParameters2R (mm)re (mm)h (mm)η ( )κ ( )β ( )Design values201.50.244.539The calculated coordinates of J on c by substituting t1 0 into Eq. (6) is set as J 2c.The calculated coordinates of J on b by substituting θ π/2 η into Eq. (15) is set asJ 2b. The calculated coordinates of I on b by substituting θ κ into Eq. (15) is set asI 2b. The calculated coordinates of I on a by substituting ϕ 0 into Eq. (19) is set asI 2a. The calculated unit tangential vector of c at J by Eq. (5) is set as τJ 2c. Thecalculated unit tangential vector of b at J by substituting θ π/2 η into Eq. (16) is setas τJ 2b. The calculated unit tangential vector of b at I by substituting θ κ into Eq. (16)is set as τI 2b. The calculated unit tangential vector of a at I by substituting ϕ 0 intoEq. (20) is set as τI 2a. The modeling parameters of one tooth are shown in Table 2 andthe result in MATLAB is shown in Fig. 9.Table 2Modeling parameters

EquationsConditionsParametersCalculated valuesEq. (6)t1 0J 2c (mm)[8.3341, 1.0504, 1.4963]TEq. (15)θ π/2 ηJ 2b (mm)[8.3341, 1.0504, 1.4963]TEq. (15)θ κI 2b (mm)[10.0000, 0.0000, 0.1177]TEq. (19)ϕ 0I 2a (mm)[10.0000, 0.0000, 0.1177]TEq. (5)\τJ 2c[-0.9924, -0.1014, -0.0698]TEq. (16)θ π/2 ητJ 2b[-0.9924, -0.1014, -0.0697]TEq. (16)θ κτI 2b[-0.0610, 0.6293, 0.7748]TEq. (20)ϕ 0τI 2a[-0.0610, 0.6293, 0.7748]TFig. 9Modeling result of the bottom edge and the conical helix edge in MATLABThe calculated and modeling results validate the G1 continuity at the two jointsand prove the accuracy of the parametric modeling method.4.2 Grinding experimentThe machining accuracy of the grinding tool path was verified by the grindingexperiment. The parameters of the grinding wheel profile and grinding process areshown in Table 3. The allowable absolute errors of machining are 0.1mm/0.5 .Table 3Parameters of the grinding wheel profile and grinding process

Grinding processRake face grindingFlank face grindingParameters of grinding wheelShapeTypeRg (mm)Hg (mm)αg ( )ν( )V-shapeCup-shape12V911V95050253245307560Based on the mathematical models of TEMC, the grinding tool path in grindingof the tested TEMC was calculated by the grinding method put forward. Part of thegrinding tool path file obtained by the grinding method put forward is shown below“.GOTO/34.687784, -1.083085, -3.482481, -0.089849, 0.769583, -0.632194GOTO/34.688929, -1.095449, -3.472860, -0.089220, 0.769622, -0.632236GOTO/34.690062, -1.107808, -3.463239, -0.088592, 0.769661, -0.632276GOTO/34.691183, -1.120164, -3.453619, -0.087964, 0.769700, -0.632316GOTO/34.692292, -1.132515, -3.444000, -0.087335, 0.769740, -0.632355GOTO/34.693389, -1.144862, -3.434381, -0.086707, 0.769780, -0.632393GOTO/34.694474, -1.157205, -3.424763, -0.086079, 0.769820, -0.632430GOTO/34.695548, -1.

Research on parametric modeling and grinding methods of bottom edge of toroid-shaped end milling cutter Lei Han1, Xuefeng Cheng1, Lei Jiang1, Rong Li1, Guofu Ding1, Shengfeng Qin2