The 14th IFToMM World Congress, Taipei, Taiwan, October 25-30, 2015 DOI Number: 10.6567/IFToMM.14TH.WC.OS6.032

Manufacture of Face-Milled Spiral Bevel Gears on a Five-Axis CNC Machine Y. P. Shih1 K. L. LAI 2 Z. H. Sun 3 National Taiwan University of Science and Technology, Taiwan Abstract: The development of CNC technology has made five-axis machines cheaper and more popular. The ability to manufacture most workpieces due to their enough degrees of freedom has driven many researchers to look for new applications on this machine. This paper presents a method for producing a face-milling cutter head and face-milled spiral bevel gears on a five-axis machine, with appropriate coordinates derived from established mathematical models and inverse kinematics. These coordinates serve as a basis for programming the NC cutting codes, whose correctness is verified using the CNC machine simulation software, VERICUT. Keywords: Face-milled spiral bevel gears, cutter head, inverse kinematics, five-axis machine.

I. Introduction Spiral bevel gears (SBGs), found extensively in gearboxes, are important elements for power transmission between intersecting axes. Two common mass production methods of SBGs in gear industry are face milling and face hobbing, both of them have similar cutting movements. In the latter, however, there is a timed relation (continuous indexing), whereas in the former, the rotation angles of the head cutter and the workpiece are independent (single indexing). The cutting tool in the face-milling process, therefore, is either a milling cutter or a grinding wheel. Until now, both these methods have been implemented using dedicated bevel gear cutting machines, which is quite cost. Modern CNC technology has led to a booming five-axis machine industry, many experts eager to find new applications for these machines. In particular, because the five-axis machine provides enough degrees of freedom to produce face-milled bevel gears, it is cheaper and offers more flexibility in automatic tool changing than a dedicated machine. Hence, many recent studies have been devoted to developing mathematical models of bevel gears produced on a five-axis machine. Most mathematical models of face-milled bevel gears, however, are derived based on the universal cradle-type bevel cutting machine. For example, Litvin et al. [1-4] established mathematical models of a face-milled bevel gear based on format, helixform, and cutter-tilt cutting systems. Fong [5] then developed a universal mathematical model of a face-milled bevel gear based on a universal hypoid generator with supplemental kinematic flank modification motions. Later, Shih et al. [6] proposed a mathematical model of both face-milled and face-hobbed bevel gears. Shih and Fong, by converting CNC machine coordinates, developed a correction method for tooth 1

[email protected] [email protected] 3 [email protected] 4 [email protected] 2

X. L. YAN 4 Sheng Yu Precision Gear Co., Ltd, Taiwan

surfaces and derived the six-axis coordinates for cutting bevel gears on a universal cradle-type bevel gear cutting machine. More recently, Wang et al. [7] simulated spiral bevel gear cutting on a general five-axis CNC machine but included no mathematical model for the cutter head. This study proposes a method for manufacturing face-milled SBGs on a five-axis machine, which begins by producing a face-milling cutter head and a face-milled SBG. After these are established, end mills are adopted to machine top, front, and side cutting faces of every blade. The coordinates for milling these faces are derived using blade geometry, while the five-axis coordinates for producing the SBG are derived from the universal machine settings using inverse kinematics. The tool used in the model is a five-axis milling machine (Quaser Machine Tools Inc., UX300 series), and model correctness is confirmed using a numerical example. II. Manufacture of the face-milling cutter head and the face-milled SBG A. Mathematical model of the face-milled SBG A1. Cutter head Face-milled SBGs are produced using a face-milling cutter head, which can be either a milling cutter or a grinding wheel (Fig. 1). Parameter rc is the nominal radius of the cutter head, r0 is the cutter radius, b is the profile angle, b is the fillet radius, H t is the height of the cutter head, and ht is the height of the cutting blade. In a milling cutter, the cutter head has a plurality of blade groups, each of which has inside and outside blades for producing the bevel gear’s convex and concave surfaces, respectively. yt

xl

yl ol

r0

zl

xn

yn

b x l



xt

ot , n

( xcf , zcf ) u

b zt , z n

ot , n

rc

xt

ht

Ht

Fig. 1 Coordinate systems of a face-milling grinding wheel

The cutting edge of the blade is a straight line with a fillet. The position of the straight-lined edge is

represented in the following equation: rt (u )  {xn cos  , xn sin  , zn ,1}   xn  r0  u sin  b  z  u cos  b  n

(1)

where u and  are the curve parameter and cutter rotation angle, and upper and lower signs refer to the inside and outside blades, respectively. A2. Universal cradle-type bevel gear cutting machine Mathematical models of an SBG are frequently derived using a virtual cradle-type bevel gear cutting machine, in which the work gear is cut by an imaginary generating gear that is rigidly connected to the cradle. The cutter head is placed on the cradle to form a tooth of the generating gear. Our proposed model uses the same generating gear to cut the pinion and gear and obtain a fully conjugated gear pair. Fully conjugated flanks, however, are not perfect for the contact condition because they do not take into account manufacturing errors and deformation under load. The machine therefore needs additional degrees of freedom to modify the tooth surface. The coordinate systems for cutting SBGs on the universal bevel gear cutting machine are as shown in Fig. 2. Coordinate systems St and S1 are fixedly connected to the cutter and work gear, and S a to S f are auxiliary coordinate systems. c and 1 are the rotation angles of the cradle and work gear. In the generating process, these rotation angles must satisfy the equation 1  Rac , where Ra is the roll ratio equal to the gear ratio of the generating gear and the work gear. This virtual machine provides nine machine settings ( i , j , S R …) for moving the cutter and work gear during the cutting process. These machine settings, which are either constant or a function of the cradle angle, can be calculated according to gear theory. yd

yc yt yb , a



y 'c

Od , c  c  c SR Em

Ob, a ,t

j i

za ,t

zb

zd ,c

xt xd

xa xb

xc

yl

ye

yf

1

B

Oe A zl

ze

Ol , f

xe

m xl , f

zf

Fig. 2 Coordinate systems for the universal cradle-type bevel gear generator [6]

The tooth surface of the work gear, obtained through coordinate transformation from coordinate system St to S1 , is represented by the following matrix equations: (2) r1(U ) (u , ,c )=M1(Uf ) (1 )M (faU ) (c )rt (u ,  )

The coordinate transformation matrices are 0 0 0 1 0 cos  -sin  0  1 1  M1(Uf ) (1 )=  0 sin 1 cos 1 0    0 0 1 0

 cos  m 0 sin  m -A  0 1 0 0  M (faU ) (1 , ,c )=  -sin  m 0 cos  m 0    0 0 1   0 0   cos ( c + c ) sin( c + c ) 1 0 0 0 1 0 E  -sin( +  ) cos( +  ) m  c c c c  0 0 1 -B   0 0   1  0 0 0 0 0 -sin j -cos j  cos j -sin j   0 0  0  0

0 S R   cos i 0 0   0 1 0  -sin i  0 1  0

0 0 0 0  1 0  0 1

0 0  0 cos i 0   0 0 1

0 sin i 1 0

A traditional bevel gear cutting machine, however, is dedicated to a specific cutting method and designed to meet its requirements. Hence, achieving flank modification motions such as cutter tilt, modified roll, or helical motion requires complicated mechanisms that are difficult to maintain. As a result, machine companies must purchase several types of bevel gear cutting machines for different cutting methods. B. Manufacture of cutter head B1. Coordinate systems for producing the cutting blades The finishing cutter head for the duplex-helical cutting method comprises a plurality of alternate inside and outside blades (see Fig. 4). As shown in Fig. 3, each blade has a side flank, top face, and front face, which form the cutting edges (CE) that cut the tooth surfaces. The side flank and top face are helical surfaces that can generate the side relief and top relief angles (  c ), while the front face controls the size of the side rake angle  c . Our model uses a table-tilting type five-axis machine to mill the blade cutting faces. This machine has three translational motions (cx , c y , cz ) and two rotational motions ( a , c ) , which latter are the table-tilting and rotation angles of the work gear, respectively. The coordinate systems for milling the three blade faces are shown in Fig 4. Coordinate systems S m and St are rigidly connected to the end mill and the cutter head, and S a to S d are auxiliary coordinate systems controlled by five coordinates. The machine constant k1 is the offsets between the work gear axis and the tilting axis, while k 2 is the distance from the table datum plane to the tilting axis. Both are measured before machining. Parameter H f is the fixture height.

c

Top face Front face

nt

Cutting edge

rt

b

c

Profile flank zm rm

ym

Fig. 3 Three faces of the cutting blade

m

Outside blade

zc

zb

yt

c

yc ob , c

a

xt

xa

cx

yb zd

xb , c

The position and unit normal vector of a selected reference point of the end mill are represented as rm  {0,   m , ht sec  b ,1} (5)  n m  {0,1, 0}

yd

where  m is the end mill radius. The upper and lower signs refer to milling the inner ( I ) and outer ( A ) sides of the blades. The position and unit normal vector of the reference point of the side flank are given by rt  {r0 cos i , r0 sin i , 0,1} (6)  n t  {cos i cos  b , sin i cos  b ,  sin  b }

c y  k1

xd Fig. 4 Coordinate systems for producing a face-milling cutter (right hand) on the five-axis machine tool

The position and unit normal vector of the end mill can be obtained in cutter head coordinate system St through coordinate transformation from S m to St : rtm  M tm ( a ,  c , cx , c y , cz )rm (3)

 n tm  L tm ( a ,  c )n m

where cosc  sin c  sin  cos c c M1t =   0 0  0  0

0 0 0 0

0 1 0 0 0 0  1 0

0 0 1 0 cos  sin  a a  0 sin a cosa  0 0 0

0 1 0 0 0 0  1 0

0 1 0 0

Fig. 5 Cutting position for the face-milling cutter blade’s side flank on the five-axis machine tool

xm

ya cz  k 2  H f

k1

ym

End mill om

ot , a

k2  H f

xm

zm

Inside blade

zt , a

Ht

nm

0 0 0   1 0 k1  0 1 (k2  H f  Ht )  0 0 1  0 0 1 0

cx  1 0  0 0  0  1  0

0 1 0 0

0 0  0 cy  k1  1 cz  k2  H f   0 1 

If the reference points of the end mill and cutter head are allowed to coincide, five coordinates can be determined by the following equations: rtm ( a ,  c , cx , c y , cz )  rt  n tm ( a ,  c )  n t

(4)

B2. Milling the side flank of the blade The side cutting edge that machines the gear tooth flank is formed by the intersection of the side flank and front face. The blade’s side flank, which dominates the angle of the cutting edge, is machined using the side of the end mill. Figure 5 shows the relative motion between the two for milling the side flank of blake.

where i is the initial setting angle of the blade, and the signs have the same referents as in the previous equation. By substituting Eq. (5) and (6) into Eq. (3), the five coordinates can be obtained as follows:  a   b ,  c  i   / 2   cx  0  c y  k y (cos  a  1)   m  A sin  a  r0 cos  a  (7)   cz  ( ht sec  b  k y sin  a  H f  k z )   A cos  a  r0 sin  a   A  H f  H t  k z The side flank is generated by screwing a line about an axis. Here, a helical ruled surface produces a side relief blade angle and guarantees the correctness of profile angle  b after blade resharpening. Rotation along the work axis ( c ), together with a proportional translation along this axis ( c y and cz ), results in a helical motion. Equation (8) shows the relation between the y, z, and c axes during milling of the side flanks.   c   hd  , L  2rc tan  c  (8) L sin  b L cos  b  , c z    c y   2 2  where hd  1 and 1 , respectively, indicate the right-hand and left-hand cutters,  is the given rotation angle, and L is a lead defined by the end relief angle  c . B3. Milling the front face of the blade The front rake angle  c is the angle between the

front face and the reference plane that passes through the cutter axis. The blade’s front faces are machined using the side of the end mill. Figure 6 shows the machining positions for the inside and outside blades. O.B.

nt

rt CE z

I.B.

x

End mill Front face CE

By substituting Eq. (12) and (13) into Eq. (3), the five coordinates can be derived as follows:  a  hd cos 1 (2 rc B ),  c    i   cx  rc  h L sin  a (14)  cy  d  A sin  a  k y (cos  a  1) 2   h L cos  a  cz  d  A cos  a  k y sin  a  H f  k z 2 

zm

x

nm

rm

ym

 m xm CE: Cutting edge Fig. 6 Cutting position for the blade’s front face

The position and unit normal vector of a selected reference point of the end mill are represented as rm  {0,  hd  m , 0,1} (9)  n m  {0, 1, 0} The position and unit normal vector of a reference point of the front face are given by  rt  {r0 cos i , r0 sin i , 0,1}  n t  {sin i cos  c  hd cos i cos  b sin  c , (10)    cos i cos  c  hd sin i cos  b sin  c ,  hd sin  b sin  c } where the upper and lower signs refer to milling the inner ( I ) and outer ( A ) sides of the blades. By substituting Eq. (9) and (10) into Eq. (3), the five coordinates can be determined as follows:  a   hd sin 1 (sin  b sin  c ),  c  tan 1 ( x, y )   cx  r0 cos(i   c )  c  k (cos   1)  h  y a d m  y   A sin  a  r0 cos  a sin(i   c ) (11)   cz  ( k y sin  a  H f  k z )   A cos  a  r0 sin  a sin(   c )   x  cos i  hd sin i cos  b tan  c   y  sin i  hd cos i cos  b tan  c B3. Milling the top face of the blade The top cutting edge, which machines the gear’s bottom surface, is formed by the intersection of the top face and front face. The top blade faces are produced using the end of the end mill. Figure 7 shows the relative position between the end mill and the work gear. The position and unit normal vector of a selected reference point of the end mill are represented as rm  {0, 0, 0,1}  n m  {0, 0,1}

 h L  rt  {rc cos(   i ), rc sin(   i ), d ,1} 2   n t  B{hd L sin(   i ),  hd L cos(   i ), 2 rc } (13)  1 B  2  L  4 2 rc2 

(12)

The position and unit normal vector of a reference point of the front face are given by

nt P2

rt P1

zm nm rm

m

ym xm

Fig. 7 Cutting position for the blade’s top face

C. Manufacture of the SBG C1. Coordinate systems for producing the SBG Theoretically, six degrees of freedom (DOF) enable relative spatial motion between two rigid bodies. However, in machining a face-milled SBG, the cutter spindle has no timed relation with the other axes, so only five DOF can satisfy the requirements of the face-milled cutting method. The coordinate systems for producing an SBG are given in Fig. 8. Coordinate systems St and S1 are fixedly connected to the cutter head and the work gear, and S a to S e are auxiliary coordinate systems that controls the relative position between the two. Parameter M d is the mounting distance of the work gear. The other parameters are as already illustrated in Fig. 4. The position of the work gear tooth surface can be expressed as Eq. (15) through coordinate transformation from the cutter coordinate system to the work gear coordinate system. r1 (u , ,c )=M1a (1 )M ae rt (u ,  )

The coordinate transformation matrices are 0 0 1 0 cos  sin  1 1 M1a =  0 sin 1 cos1  0 0 0

0 0 0  1

(15)

0 0 Md  H f  k2   1 0 0   0 1 k1  0 0 1  0 cosa 0 1 0 0 0  1 0 0 0 1 0 Cx  0 sin a 0 0 0 1 0    0 0 1 0 0 0 1  (Cy  k1 )  cos b  sin b 0 0   sin  cos  0 0 0 b b   (Cz  H f  k2 )  0 0 1 0   1 0 0 1  0

0 0 1 0  cos   sin  c c Mae =  0 sin c  cos c  0 0 0  sin a  0   cosa   0 1 0 0 0 1 0  0 0 1  0 0 0

0 1 0 0 0 0  1 0

Fig. 8 Coordinate systems for bevel gear cutting on the five-axis machine tool

C2. Derivation of the five coordinates for bevel gear cutting

Regardless of whether the work gear is produced on a cradle-type or a five-axis machine, the relative positions of the cutter with respect to the work gear should be identical. Comparing Eq. (2) with Eq. (15) thus satisfies the following relation:  e11 e12 e13 e14  e e e23 e24  (16) M ae =M (faU ) (c )   21 22  e31 e32 e33 e34    0 0 1 0 The elements on the right side of the equation are known as the functions of cradle angle c . By comparing the upper 3  3 submatrix of Eq. (16), three rotational angles can be derived as Eq. (17)  a (c )   sin 1 e13  1 1 (17)  b (c )  tan ( x, y )  tan (e11 , e12 )  1  c (c )  tan (e33 , e23 ) The translation vectors (fourth column) in Eq. (16) are equal, yielding three equations to solve the translational positions:

Cx (c )  e24 cos c  e34 sin c  C y (c )  D sin a  E cos a  k1  Cz (c )  E sin a  D cos a  H f  k2   D  e14  H f  k2  M d  E  e sin   e cos   k 24 c 34 c 1 

(18)

Because the cutter axis is a spindle, angle b has no influence on the machined gear’s tooth surface geometry. The rotation angle c , which includes the generating angle and the incremental angle of work gear, can be expressed as follows: c (c )  1  c  Rac  c (c ) (19) Because the five coordinates for the cutting positions are functions of the cradle angle c , these position functions can be approximated by a Taylor series expressed as Eq. (20). Here, the Taylor series (also called a Maclaurin series) is centered at zero. Degree six is assumed to meet the requirement of machining accuracy. n f ( i ) (0)   i 6  p c  Rp( n ) (c )   aic i  f p (c )   (20) i!  i 0 i 0  a  c , c , c , , x y z a c  D. Numerical examples Our example is a face-milled SBG pair cut by the duplex-helical method, which adopts a cutter hand with alternate inside and outside blades to produce both flanks in one operation. This method is well-known for its high productivity. The basic design parameters for the example gear pair and the parameters for the five-axis machine are listed in Table 1. The design parameters for the cutter head are given in Table 2, in which the profile angle and cutter radius are calculated according to reference [8], and other parameters are set based on cutter design experience. In this paper, the values of most parameters are provided using the SI unit system. The three cutting faces of the cutter head are machined using end mills. Substituting the cutter head-related parameters into Eqs. (7), (11), and (14) yields the five coordinates for milling each face of the cutter blade (see Table 3). Given  , the adequate angle for milling a complete side flank, three incremental coordinates can be obtained for a helical motion. Figure 9 shows the cutting position for milling the side flanks of the cutter head on the five-axis machine. TABLE 1 Basic parameters of the numerical gear pair

Items (A) Basic gear data Number of teeth Outer module Pressure angle Spiral angle (B) Gear blank data Pitch angle Face angle Outer diameter

Pinion

z met

Ring gear

16

33 3.500

n m

35 L.H.

35 R.H.

 a

25.866 30.727

64.133 66.464

d ae

63.408

117.221

20.000

Ring gear 6.869 19.500 62.000 38.000

Items he Outer whole depth Face width b Md Mounting distance (C) Assembly data Shaft angle  V Offset Axial setting H (E) Five-axis machine tool k1 Offset along yb (Fig. 4)

Pinion

90.000 0.000 0.000

0.000 0.000 -0.022

Offset along za (Fig. 4)

k2

0.085

Fixture height for cutter

Hf

88.820

Fixture height for gear

Hf

150.000

97.000

TABLE 2 Parameters of the cutter head and the end mill

Inside blade I A

Outside blade I A

(A) Cutter head Number of blades

zn

10

10

Cutting direction

hd

R. H.

R. H.

b 24.000 13.000 16.000 21.000 Profile angle Nominal radius of cutter r 55.000 55.000 c head r0 54.250 55.550 54.450 55.750 Cutter radius b Fillet radius 0.600 0.600  0 -18.000 Initial setting angle i hi 12.500 12.500 Blade height H 42.500 42.500 Cutter height i c 12.245 12.245 Top relief angle Front rake angle (B) End mill Dia. for the side flank Dia. for the front face Dia. for the top face

c

20.000

20.000

m m m

5.000

5.000

3.000

3.000

3.000

3.000

Side flank

a

I I.B. A O.B I . A Front face I.B. O.B. Top face P1 P2

c

Coordinates cx cy

-24.000 -90.000 13.000 -90.000 -21.000 -108.000 16.000 -108.000 -7.996 -18.392 -5.410 1.283 12.245 0.000 12.245 -13.000

0.000 0.000 0.000 0.000 51.479 52.622 55.000 55.000

39.529 13.809 39.902 9.042 43.604 40.179 39.512 36.866



c

15.000 14.500 11.000 15.000

-15.000 -14.500 -11.000 -15.000

c y 1.271 -0.680 0.821 -0.861

Items Tilt angle Swivel angle Initial cradle angle setting

i j c

Pinion 1.106 28.332 64.598

Gear 0.995 -67.396 -63.349

SR

50.171

50.234

Vertical offset Increment of machine center A to back Sliding base feed setting B m Machine root angle Ra Roll ratio

-0.236

0.000

-0.332

-0.459

0.965 -2.910 c

2.399

cz -2.855 -2.943 -2.139 -3.004

The universal machine settings for producing the pinion and the gear are determined according to reference

21.339

58.508

2.27513

1.10112

TABLE 6 Cutting position for generating the pinion

Coord.

c (rad)

a c

TABLE 4 Parameters of helical motion for milling the side flanks

Items I I.B. A I O.B. A

TABLE 5 Universal machine settings

Em

cz

-8.853 88.667 -1.227 94.787 1.682 -27.709 27.852 27.278

[8] and shown in Table 5. Substituting machine settings, machine constants, cutter height, and mounting distance of the work gear into Eqs. (17) to (19) yields the five coordinates for producing the work gears. Table 6, however, only displays the coordinates for the pinion, which are functions of the cradle angle. A cradle angle c of between 0.2734 and −0.2959 rad enables a generating motion from the toe to the heel of work gear. Figure 10 shows the cutting position for producing the pinion on the five-axis machine.

Radial setting

TABLE 3 Cutting position for the cutter head

Items

Fig. 9. Cutting position for milling the cutter head’s side flank on the five-axis machine

cx cy

cz

Kinematic positions 0.2734  c  0.2959

68.0091+0.8960c  0.3258c 2  0.1514c 3 +0.0268c 4 +0.0080c 5  0.0008c 6 0.9620  131.0550c  0.4875c 2 +0.1151c 3 +0.0428c 4  0.0054c 5  0.0017c 6 45.4031  21.9571c +22.7437c 2 +3.8708c 3 1.9085c 4  0.2360c 5 +0.0638c 6 174.5600  44.1593c  11.6251c 2 +7.3765c 3 +1.0697c 4  0.3679c 5  0.0499c 6 69.3842+0.1589c  1.1282c 2  0.5113c 3 +0.0940c 4 +0.0256c 5  0.0031c 6

head and SBG are established, inverse kinematics can be used to derive the five-axis machine coordinates for their production. These coordinates can then be used to program the NC codes. Finally, the correctness of the coordinates can be confirmed using the VERICUT CNC machine simulation software. IV. Acknowledgments The authors are grateful to the R.O.C.’s Ministry of Science and Technology for its financial support. Part of this work was performed under Contract No. MOST 1032221-E-011-026.

Fig. 10. Cutting position for cutting the pinion when c  0 on the five-axis machine

E. Simulation result using VERICUT The coordinates listed in Tables 3 and 4 are adopted to program the NC codes for milling the cutter blade faces. Figure 11 illustrates the VERICUT simulation of the cutter blade milling. The finished cutter head is then compared with the theoretical one to demonstrate the accuracy of the derived coordinates. The NC codes for producing the pinion are programmed based on the position functions listed in Table 5. The machining consists of two operations: a formate motion enabled by setting c  0 and a generating motion enabled by a range of c . The simulated pinion cutting is illustrated in Fig. 12, and the result, exported as an STL (STereoLithography) file, is shown in Fig. 13. The simulated and theoretical tooth surfaces are then compared to reveal the flank topographic deviations. As Fig. 14 shows, the thickness error is zero, and the sum of squared errors is only 25,504  m 2 , which could be caused by simulation software error. This comparison of the tooth surface produced by the five-axis versus the universal machine thus confirms the accuracy of the mathematical model and the efficacy of the proposed method.

Fig. 12 VERICUT simulation of the pinion cutting.

Fig. 13. Finished pinion (STL file).

Fig. 11. VERICUT simulation of cutting head production.

III. Conclusions A five-axis machine has enough degrees of freedom to allow implementation of face-milling cutting for SBGs. This paper successfully develops a model for manufacturing face-milled SBGs on the five-axis machine. Once the mathematical models of the face-milling cutter

Fig. 14. Flank topographic deviations between the simulated and theoretic tooth surfaces of the pinion.

References [1] Litvin F. L. and Gutman Y. Methods of Synthesis and Analysis for Hypoid Gear-Drives of “Formate” and “Helixform”—Part 1. Calculations for Machine Settings for Member Gear Manufacture of the Formate and Helixform Hypoid Gears. Journal of Mechanical Design, vol. 103, pp. 83-88, 1981. [2] Litvin F. L. and Gutman Y. Methods of Synthesis and Analysis for Hypoid Gear-Drives of “Formate” and “Helixform”—Part 2. Machine Setting Calculations for the Pinions of Formate and Helixform Gears. Journal of Mechanical Design, vol. 103, pp. 89-101, 1981. [3] Litvin F. L. and Gutman Y. Methods of Synthesis and Analysis for Hypoid Gear-Drives of “Formate” and “Helixform”—Part 3. Analysis and Optimal Synthesis Methods for Mismatch Gearing and its Application for Hypoid Gears of “Formate” and “Helixform”. Journal of Mechanical Design, vol. 103, pp. 102-110, 1981. [4] Litvin F. L., Zhang Y., Lundy M., and Heine C. Determination of Settings of a Tilted Head Cutter for Generation of Hypoid and Spiral Bevel Gears. Journal of Mechanisms, Transmissions, and Automation in Design, vol. 110, pp. 495-500, 1988. [5] Fong Z. H. Mathematical Model of Universal Hypoid Generator with Supplemental Kinematic Flank Correction Motions. Journal of Mechanical Design, vol. 122, pp. 136-142, 2000. [6] Shih Y. P., Fong Z. H., and Lin G. C. Y. Mathematical Model for a Universal Face Hobbing Hypoid Gear Generator. Journal of Mechanical Design, vol. 129, pp. 38-47, 2007. [7] Wang Y., Zhao L., and Kou J. B. Virtual Machining of Spiral Bevel Gear Based on VERICUT. Advanced Materials Research, pp. 1885-1889, 2011. [8] Gleason Works, Calculation Instructions: Generated Spiral Bevel Gears, Duplex-Helical Method, Including Grinding, Rochester, NY, USA, 1971.