An innovative approach to NC programming for accurate five-axis flank milling of spiral bevel or hypoid gears

Accepted Manuscript An innovative approach to NC programming for accurate five-axis flank milling of spiral bevel or hypoid gears Yuansheng Zhou, Zezhong C. Chen, Jinyuan Tang, Shengjun Liu PII: DOI: Reference:

S0010-4485(16)30145-2 http://dx.doi.org/10.1016/j.cad.2016.11.003 JCAD 2485

To appear in:

Computer-Aided Design

Received date: 7 March 2016 Accepted date: 15 November 2016 Please cite this article as: Zhou Y, Chen ZC, Tang J, Liu S. An innovative approach to NC programming for accurate five-axis flank milling of spiral bevel or hypoid gears. Computer-Aided Design (2016), http://dx.doi.org/10.1016/j.cad.2016.11.003 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

*Highlights (for review)

Highlights 

Five-axis flank milling is applied to manufacture spiral bevel or hypoid gears



Tangency is formulated for cutter envelope and designed surfaces along a given line



An closed-form approximate model is proposed to represent geometric deviations



Geometric deviations are minimized to contact area rather than whole tooth surface

*Manuscript Click here to view linked References

An innovative approach to NC programming for accurate five-axis flank milling of spiral bevel or hypoid gears Yuansheng Zhoua , Zezhong C. Chenb , Jinyuan Tanga,∗, Shengjun Liuc a State

Key Laboratory of High Performance Complex Manufacturing, Central South University, Changsha, Hunan, China, 410083 b Department of Mechanical and Industrial Engineering, Concordia University, Montreal, Canada, H3G1M8 c School of Mathematics and Statistics, Central South University, Changsha, Hunan, China, 410083

Abstract To transfer power, a pair of spiral bevel or hypoid gears engages. From beginning to end of two tooth surfaces engaging with each other: for their rigid property, they contact at different points; and for their plastic property, they contact at small ellipses around the points. On each surface, the contact line (or called as contact path) by connecting these points and the contact area by joining these ellipses are critical to driving performance. Therefore, to machine these surfaces, it is important to machine the contact line and area with higher accuracy than other areas. Five-axis flank milling is efficient and is widely used in industry. However, tool paths for flank milling the gears, which are generated with the existing methods, can cause overcuts on the contact area with large machining errors. To overcome this problem, an innovative approach to NC programming for accurate and efficient five-axis flank milling of spiral bevel or hypoid gears is proposed. First, the necessary conditions of the cutter envelope surface tangent with the designed surface along a designed line are derived to address the overcut problem of five-axis milling. Second, the tooth surface including the contact line and area are represented using their machining and meshing models. Third, according to the tooth surface model, an optimization method based on the necessary conditions is proposed to plan the cutter location and orientation for flank milling the tooth surface. By using these planned tool paths, the overcut problem is eliminated and the machining errors of contact area are reduced. The proposed approach can significantly promote flank milling in the gear manufacturing industry. Keywords: Five-axis flank milling; tool path planning; envelop surface; geometric deviations; spiral bevel and hypoid gears; tooth surface

1

Nomenclature α

Profile angle to define the cutter shape

mbc

Roll ratio

Sr

Radial distance

q2

Gear cradle angle

∆XB

Sliding base

∗ Corresponding

author Email addresses: [email protected] (Yuansheng Zhou), [email protected] (Zezhong C. Chen), [email protected] (Jinyuan Tang), [email protected] (Shengjun Liu)

Preprint submitted to Computer-Aided Design

October 8, 2016

2

∆Em

Blank offset

∆XD

Machine center to back

γm

Machine root angle

αg

Pressure angle of blade cutting edges

rf

Fillet radius of blade cutting edges

Ru

Average radius

Pw

Point width

c

Clearance

Γg

Pitch angle

Aq

Pith cone distance

hq

Tooth depth

bqg

Gear dedendum

µ

Angle parameter to define the tool axis

ε

Approximate model of geometric deviation

κ

Normal curvature

1. Introduction

3

Spiral bevel and hypoid gears are critical components of power transmission systems of automobiles, helicopters,

4

and power generators etc. The axes of a pair of spiral bevel gears intersect with each other, while the axes of a pair of

5

hypoid gears skew with each other. Currently, they are mainly produced in face milling, face hobbing and hobbing on

6

special machine tools, such as Gleason, Oerlikon, and Klingelnberg machines. These machining strategies are quite

7

efficient and cost effective for a large volume of gears. However, the investment of such gear machining is enormous

8

and prohibitive for small businesses, which normally produce these gears in small batch. As an emerging alternative,

9

flank milling of these gears on five-axis CNC machining centers is in favor by small gear manufacturers and worn

10

gear re-manufacturers. Flank milling is quite efficient in removing excessive stock material with the tool’s long side

11

cutting edges and it has been widely employed in industry [1, 2]. However, it has rarely been applied to spiral bevel

12

and hypoid gears machining. Now, some companies, such as DMG and Gleason, start to adopt five-axis flank milling

13

in spiral bevel and hypoid gears production. Unfortunately, their gears accuracy is not high and the machining time is

14

relatively long. To develop this technique, first, it is necessary to know the characteristics of the gearing mechanism.

15

To transfer power, a pair of spiral bevel or hypoid gears engage. From beginning to end of two tooth surfaces

16

engaging with each other: for their rigid property, they contact at different points [3, 4]; and for their elastic property,

17

they contact at small ellipses around the points. On each surface, the contact line by connecting these points and

18

the contact area by joining these ellipses are critical to driving performance. The gear and pinion surfaces including

19

the contact line and area can be accurately represented using their machining and meshing models. Therefore, to

20

machine these surfaces, it is important to machine the contact line and area with higher accuracy than other areas. For 2

21

this purpose, it is crucial to represent these gears tooth surfaces and the contact line and area based on the current

22

machining approaches.

23

Litvin et al. [3–5] established the kinematic chain in face milling spiral bevel gears and modeled their tooth surfaces.

24

Similar researches were conducted for face milling [6, 7], face hobbing [8, 9], and hobbing [10] of spiral bevel and

25

hypoid gears. With the tooth surfaces models, the tooth contact analysis (TCA) is conducted to obtain the transmission

26

errors and contact conditions. In case of inappropriate results , the tooth surfaces should be modified by changing the

27

machine setting or cutter motion. Litvin et al. [3–5] proposed the local synthesis method to determine the machine

28

setting for pinion machining, as a result, a desired contact ellipse was achieved at the pinions mean point and the

29

transmission error is controlled. Achtmann and B¨ar [11] applied modified helical motion and roll to produce optimally

30

fitted bearing ellipses. Fan [12] used higher order polynomials to represent the cradle increment angle of the machine

31

setting than the conventional way, and developed TCA programs in the Gleason commercial software, CAGETM .

32

Later, Fan [13] proposed a generic tooth surface model and an enhanced algorithm to simulate the tooth surfaces

33

contact. Simon [14] reduced the transmission errors by defining the cradle radial setting and the cutting ratio with fifth-

34

order polynomial functions and optimizing them. In the work [15], these setting and ratio were applied on a hypoid

35

generating machine. Furthermore, an algorithm was developed to ensure the relationship between the machine setting

36

of the CNC hypoid generator and that of the cradle-type generator [16].

37

In terms of CNC milling of spiral bevel and hypoid gears, Suh et al. [17] machined the tooth surfaces of spiral bevel

38

gears in four-axis or 3/4-axis CNC end milling. Later, Suh et al. [18] cut the spiral bevel gears with crown on four-axis

39

CNC milling machines. Alves et al. [19] machined the spiral bevel gears in five-axis CNC end milling. Zhou [20]

40

adopted five-axis flank milling to cut the spiral bevel gears with a ruled tooth surface design. Since the tool paths were

41

planned with the previous methods of five-axis flank milling, the contact area were overcuts. To better understand tool

42

path planning for flank milling, the literature is reviewed.

43

Tool path planning for five-axis flank milling is to calculate the tool locations and orientations so that the machined

44

surface is within the tolerance of the designed surface. The machined surface is calculated as part of the cutter envelope

45

surface moving along the planned tool path. The deviations between the machined surface and designed surface are

46

named geometric deviations (or geometrical deviations) [21–24]. Lartigue et al. [21] optimized the tool paths in five-

47

axis flank milling to minimize the geometric deviation between the cutter envelope and the design surface. Pechard

48

et al. [23] minimized the geometric deviation while preserving the cutter trajectory smooth in five-axis high-speed

49

flank milling. Zhou et al. [24] used a geometric envelope approach to calculate the accurate geometric deviations for

50

a specific CNC machined tool. Bedi et al. [25] proposed a tool path planning method by keeping the cutter tangent to

51

two curves during flank milling. Menzel et al. [26] calculated the tool paths by positioning the flat end mill tangential

52

to two guiding rails and one rule line. Li et al. [27] compared three different methods by calculating their geometric

53

deviations caused in flank milling. Chiou [28] determined the cutter positions for five-axis ruled surface machining

54

by comparing the swept profile with the ruled surface. Senatore et al. [29] studied influence of cutter axis adjustment,

55

and then used it to reduce the geometric deviation in flank milling of ruled surfaces. Chu and Chen [30] introduced

56

the developable surface approximation to generate the interference free tool paths for five axis flank milling of ruled

3

57

surfaces. Chu et al. [31] considered the interpolation sampling time of the CNC controller in NC tool path planning to

58

improve machining accuracy of five-axis flank milling. Wu et al. [32] optimized tool paths of five-axis flank milling of

59

ruled surfaces based on the dynamic programming techniques. Hsieh and Chu [33] applied the PSO-based optimization

60

scheme to minimize the geometric deviations and the graphics processing unit method to reduce the computation time.

61

Chaves-Jacob et al. [34] reduced tool interference in five-axis flank milling of free-form surfaces by optimizing the

62

tool shape. Zheng et al. [35] generated the tool paths for flank milling centrifugal impellers by considering cutter size

63

and interference. Zhu et al. [36] simultaneously optimized the cutter shape and tool path for five-axis flank milling.

64

Zhu et al. [22] calculated the geometric deviation based on the envelope theory of sphere congruence. Li and Zhu [37]

65

considered the cutter run-out effect into the envelope surface modeling and the tool path optimization for five-axis flank

66

milling with a conical cutter. Zhu and Lu [38] proposed the necessary and sufficient conditions for tangent continuity

67

of the swept tool envelopes of flank milling. Zhu et al. [39] optimized tool paths of five-axis flank milling globally to

68

satisfy the minimum zone criterion recommended by the ANSI and ISO standards. Gong et al. [40] applied the three

69

points offset strategy to optimize tool paths for flank milling ruled surfaces with cylindrical cutters. Subsequently,

70

Gong and Wang extended the work [40] to a global optimization [41], and they generated a tool path for flank milling

71

free-form surfaces with an approximate model of the envelope surface of a generic cutter. Furthermore, Gong and

72

Wang [42] proposed a tool path generation method of flank milling considering constraints for ball-end cutters. Harik

73

et al. [43] gave a detailed review about the cutter trajectory optimization related to five-axis flank milling. Concave side of tooth suface

Convex side of tooth suface Cutter

Toe P

hq

l

Heel oc

Contact line (a) Contact line on the tooth surface

q Tooth surface

(b) A tool path planning strategy

Figure 1: Five-axis flank milling the tooth surfaces with an existing tool path planning strategy

74

To machine surfaces in flank milling, the aforementioned methods focus on planing tool paths to minimize the

75

overall geometric deviations of designed surfaces. The tooth surfaces of the spiral bevel and hypoid gears are different

76

from the conventional surfaces. The contact line and area of a tooth surface is very important and should be machined

77

with higher accuracy than other areas. Moreover, according to [20], overcuts are happened on the contact line and area

78

while a tool path planning strategy, which is widely used in the existing methods, is applied to flank mill tooth surfaces.

79

To determine the tool tip point oc and tool axis l , several points of this strategy are stated as follows.

80

• The cutter surface is planned to tangent with the tooth surface along the contact line, which is shown in Fig. 1 (a).

81

For a point q on the contact line, the tangent points on the designed surface and cutter surface are q and p,

82

respectively, as shown in Fig. 1 (b). p is named cutter contact (CC) point.

83

• Because the space of each tooth slot becomes narrower from the top to bottom, a conical cutter is used to machine 4

84

85 86

the tooth surface. • Since the depth of the tooth surface varies from one end (toe) to the other end (heel), hq is varied along the contact line in order to use one path to machine the convex or concave side of the tooth surface.

87

While this example is considered as a general case in five-axis flank milling, the contact line and tooth surface can be

88

treated as the designed line and designed surface, respectively. Theoretically, if the designed surface is tangent with

89

cutter envelope surface along the designed line, there are no overcuts or undercuts happened on the designed line. This

90

could be satisfied with two necessary conditions: (1) the first item in the aforementioned strategy is ensured; and (2)

91

CC point p is also contributed as a point on the envelope surface. These necessary conditions are formulated in Section

92

2, and they can be used to explain the overcuts happened in the aforementioned example.

93

In this paper, the necessary conditions of the cutter envelope surface tangent with the designed surface along a

94

designed line are proposed in Section 2. The tooth surface, which is the designed surface of five-axis flank milling, is

95

given in Section 3, and the contact line and area are also illustrated. In Section 4, an optimal tool path is proposed to

96

reduce the geometric deviations of the contact area based on the necessary conditions. Since the geometric deviation

97

is usually calculated with numerical methods [22–24], it makes the further calculation very complicated. To overcome

98

this problem, a closed-form approximate model of the geometric deviation is proposed. Subsequently, an optimal tool

99

path is obtained to minimize the geometric deviations of the contact area. An example is given to validate the proposed

100

method in Section 5. Conclusions are given in Section 6.

101

2. The necessary conditions of the envelope surface tangent with the designed surface along a designed line in

102

five-axis milling

103

The envelope surface is simply introduced in Subsection 2.1 based on previous literatures [3, 7]. Subsequently,

104

the necessary conditions of the envelope surface tangent with the designed surface along a designed line in five-axis

105

milling, is newly derived and illustrated in Subsection 2.2. It is worthwhile to mention that this theory can be applied

106

to both five-axis flank milling and end milling. Hence we use five-axis milling rather than five-axis flank milling in

107

Section 2.

108

2.1. Formulations of the envelope surface of a surface of revolution

109

As shown in Fig. 2, a surface of revolution is the surface formed by rotating a planar generatrix around an axis l

110

which lies in the same plane as the generatrix. A point p on the surface of revolution can be represented as a vector

111

form p(h, θ), where h and θ are two independent parameters as shown in Fig. 2. When the surface of revolution

112

moves continuously under a general motion defined with parameter φ, a family surfaces of the surface of revolution

113

is generated in this process, which is mentioned as generating process. The boundaries of the family surfaces are the

114

envelope surfaces.

115

According to the meshing theory [3, 4], or envelope theory [44], the envelope surface and the surface of revolution

116

are tangent with each other along a curve at every instant, and this curve is regarded as grazing curve (also referred

5

n

θ

p

α

ph

ρ l

Generatrix h

oc

Figure 2: A general surface of revolution.

117

to as contact curve, generating curve, etc). The point on the grazing curve is called as grazing point, which can be

118

calculated according to the tangency condition. Subsequently, the envelope surface of a surface of revolution can be

119

calculated as a closed-form result according to the geometric meshing theory (or geometric envelope approach) [7, 24]

120

as

121

r(h, φ) = oc (φ) + h(φ) · l(φ) + ρ(h) · n(h, φ)

122

where oc is the tip point of the surface of revolution; n is the unit normal of the surface of revolution at point p; ρ is

123

the distance between p and ph , which is the intersection point of n and l. The angle formed by n and l is represented

124

as α. n in Eq. (1) can be calculated as as [7, 24]

125

n(h, φ) =

cos α · vh2 2 (l × vh )

·l−

(l · vh ) · cos α (l × vh )

2

· vh ±

q 2 (l × vh ) − cos2 α · vh2 (l × vh )

2

(1)

· (l × vh ) .

(2)

126

The normal curvature of the envelope surface is needed for further calculations. According to Eqs. (1) and (2), the

127

closed-form representation of the normal curvature is obtained in Appendix. The detail about this calculation process

128

can be referred to [7].

129

2.2. The necessary conditions

130

As shown in Fig. 3, a generating process is considered to generate an envelope surface tangent to a designed surface

131

along a designed line, which lies on the designed surface. The cutter is represented as a surface of revolution. Assume

132

that the designed surface is fixed and the surface of revolution moves relative to the designed surface with a motion

133

defined by parameter φ. At each instant of the generating process, the surface of revolution is tangent to the designed

134

surface at a point, which is represented on both surface as p and q, respectively (The tangency condition could be

135

happened at more points, even a line. Here we only discuss the circumstance of a point as the example). As shown

6

Designed line q(ϕ) CC line p(ϕ) ph

ρq nq q

p

l

hq

oc

Figure 3: Formulations of the necessary conditions

136

in Fig. 3, the trajectories of the tangent point on both surfaces are the CC line and designed line, respectively. nq is

137

the unit normal of both surface at the tangent point. ph is the intersection point of nq with the axis l of the surface of

138

revolution. hq is the distance from the tip point oc to ph . It worths mentioning that hq could be varied and represented

139

as a function of φ during the generating process. Since the shape parameter ρq is a function of hq , ρq can be treated as

140

a function of φ. Then we have ph (φ) = q(φ) + ρq (φ) · nq (φ).

141

142

According to the derivative of Eq. (3) with respect to φ, the velocity of ph is calculated as vph (φ) =

143

144

147

dq(φ) dρq (φ) dhq (φ) dnq (φ) + · nq (φ) + ρq (φ) · . · dφ dhq dφ dφ

(4)

By multiplying nq (φ) for both side of Eq. (4), we have nq (φ) · vph (φ) =

145

146

(3)

dρq (hq ) dhq (φ) · . dhq dφ

(5)

According to [7, 24], nq (φ) · vph (φ) = 0 is the necessary condition of that the CC point is a grazing point. Then we have

148

dρq (hq ) dhq (φ) · = 0. dhq dφ

149

The lefe side of Eq. (6) is directly determined by the CC line on the cutter surface. To satisfy Eq. (6), different cases

150

are stated as follows.

151

• CC line is a point on the surface of revolution. hq (φ) and ρq (φ) are two constants.

152

•

dhq (φ) dφ

(6)

= 0. hq (φ) is a constant during the generating process. Subsequently, ph is the same point on the axis of

7

153

the surface of revolution; ρq (φ) is a constant; CC line on the surface of revolution lies on a circle, which is the

154

intersection between the surface of revolution and the plane passing through p and perpendicular to l. •

155

dρq (hq ) dhq

= 0. ρq (φ) is a constant during the generating process. An example of this case is that the surface of

156

revolution is a cylinder. Subsequently, hq (φ) could be varied and the CC line could be any curve on the surface

157

of revolution that passes through p.

158

For five-axis milling, the necessary conditions of the envelope surface tangent with the designed surface along a de-

159

signed line are summarized as two aspects.

160

• The cutter surface is tangent to the designed surface along the designed line.

161

• The CC line on the cutter surface should be satisfied to Eq. (6).

162

In the previous literatures of five-axis CNC milling, the first item has been well applied, but not the second item [45].

163

The first item ensure the tangency condition between the cutter and the designed surface. However, the tangency

164

condition between the cutter envelope surface and the designed can not be guaranteed by only considering the first

165

item, but it does by applying both items. With the necessary conditions, the overcuts occurred in [20] can be explained. For this case, since

166

dρq (hq ) dhq

dhq (φ) dφ

6= 0

6= 0, Eq. (6) is not satisfied. Hence, overcuts are occurred. This example also shows that our proposed

167

and

168

necessary conditions are important supplements to the previous knowledge of five-axis CNC milling.

169

3. Representations of the tooth surface and its contact line and area

170

Since the tooth surface of spiral bevel gears is barely introduced in the previous literatures of five-axis flank milling,

171

it is necessary to give a brief introduction here. The tooth surface manufactured with conventional approaches are

172

directly related to manufacture approaches, here we use the face milling method as the example to model the tooth

173

surface, which is used as the designed surface in five-axis flank milling. Subsequently, the contact line and area are

174

also introduced.

175

3.1. The generation of face-milled generated gears

176

The face milling method includes two types, generated method and non-generated method. They have different

177

motions in the process of cutting each tooth slot. As shown in Fig 4, three rotation motions marked as (1) ∼ (3)

178

are rotations of the head-cutter, cradle, and gear blank, respectively. During the process of cutting each slot, the gear

179

blank rotation is applied for the generated method but not for the non-generated method. The other two rotations

180

are applied on both methods. Subsequently, the non-generated method can be treated as a specific case of generated

181

method. Hence, we will focus on the generated method. The head-cutter rotation provides the cutting velocity. The

182

cradle rotation and gear rotation are related as a roll function as

183

φb (φc ) = mbc · φc − C · φ2c − D · φ3c − E · φ4c − F · φ5c 8




(7)

(1)

Cradle Head-cutter

yg

zm,zc

Blades

om,oc ym

(2)

xg

ym xc q ϕc yc om,oc xm og

xm

om

on

zg

zm

xb ϕ b γm ϕb

oa

ob

(3) zb

yb

Gear blank

Figure 4: Generation motions and kinematics of face-milled generated gears

184

where φb and φc are the rotation angles of the gear blank and cradle, respectively; mbc is roll ratio; C, D, E, F are

185

the modified roll coefficients.

186

Fig. 4 is an example of face milling on a Gleason’s CNC machine with the generated method. The installment of the

187

cradle is determined by two machine-settings, radial distance Sr = koc og and cradle angle q2 . The settings of the gear

188 189

blank is represented by other four parameters, and they are the sliding base ∆XB = kom on k, the blank offset ∆Em = kon oa k, the machine center to back ∆XD = koa ob k and the machine root angle γm . With these machine-settings, the

190

kinematic relation between the head-cutter and the gear blank is determined. The homogeneous transformation matrix

191

(HTM) from the head-cutter coordinate system Sg to gear blank coordinate system Sb is calculated as

192

193 194

195



  Mbg (φc ) = RZ (−φb ) · T  

0 0 −∆XD





0

   π     · RY −( − γm ) · T  ∆Em 2   −∆XB





Sr · cos q2

     · RZ (φc ) · T  Sr · sin q2   0



   (8) 

where R and T are rotation and translation HTMs, respectively. For example, RZ (−φb ) is the rotation HTM by rotat  T T ing along Z axis with an angle −φb , and T [0, 0, − ∆XD ] is the translation HTM with a vector [0, 0, − ∆XD ] . 3.2. Representation of the head-cutter surfaces

196

With head-cutter rotation, the head-cutter surfaces are two surfaces of revolution formed by blade cutting edges. In

197

this example, the blade cutting edges on both concave and convex sides are a straight line with circular fillet, as shown

198

in Fig. 5. The segment of the straight line with the profile angle αg generates the working part of the gear tooth surface.

199

The circular arc of radius rf generates the fillet of the tooth surface. Subsequently, the working part of head-cutter 9

concave side ph θ

convex side n straight line Xw

αg

h

φ

circular fillet xg

og yg

n

rf Pw

Ru

zg, lg Figure 5: Blade cutting edges

200

surface can be represented as p(h, θ) = og + h · lg + ρ(h) · n(h, θ)

201

202

where ρ(h) =

203

α(h) =



Ru ±

Pw ± h · tan αg 2

π ∓ αg . 2



(9)

· csc(α(h))

204

Ru and Pw are the average radius and point width, respectively, as shown in Fig. 5. The upper and lower sign refer to

205

the concave and convex side, respectively. Similarly, the fillet of the head-cutter surface can be represented as  π i p(ϕ, θ) = og + h(ϕ) · lg + ρ(ϕ) · n(ϕ, θ), ϕ ⊆ 0, − αg 2

206

207

208

209

where

(10)

  Pw h(ϕ) = rf − Xw · cot ϕ, Xw = Ru ± ∓ rf · (sec αg − tan αg ) 2 hπ π i ρ(ϕ) = (Xw ± rf · sin ϕ) · csc ± (ϕ − ) . 2 2

3.3. Formulation of the tooth surface

210

According to Eqs. (1) and (2), if the head-cutter parameters are given, we need to calculate og (φc ), lg (φc ),

211

vh (h, φc ) and vh (ϕ, φc ) to obtain the closed-form expression of the tooth surface. vh (h, φc ) and vh (ϕ, φc ) are calcu-

212

lated for the working part and fillet of the tooth surface, respectively. In head-cutter coordinate system Sg , we have

213

h ogg = 0

0

iT h g 0 , lg = 0

10

0

iT 1

(11)

214

where ogg is the vector og expressed in Sg , and this illustration is also applied to other similar items. Subsequently, they

215

can be calculated in the gear blank coordinate system Sb as     h obg (φc ) og   = Mbg (φc ) ·  g  = M14 1 1     h lb (φ ) lg  g c  = Mbg (φc ) ·  g  = M13 0 0

216

217 218

M24

M34

iT 1

(12)

M23

M33

iT 0

(13)

219

where Mij is the element at the ith row and j th column of Mbg . With Eqs. (12) and (13), vh (h, φc ) can be calculate

220

in Sb as

221

vhb (h, φc ) =

 dobg (φc ) dlbg (φc ) dM14 +h· = dφc dφc dφc

dM24 dφc

dM34 dφc

T

+h·



dM13 dφc

dM23 dφc

dM33 dφc

T

.

(14)

222

vhb (h, φc ) is used to calculate the envelope surface of the working part. Similarly, vhb (ϕ, φc ) can also be calculated and

223

it is used to calculate the envelope surface of the fillet. Subsequently, the envelope surface of the head cutter surface

224

can be obtained by submitting Eqs. (12) and (13) into Eqs. (1) and (2).

225 226

The tooth surface is obtained as the intersection of the envelope surface and gear blank. An example is given with the data in Tab. 1, the tooth surface of the spiral bevel gear is obtained as shown in Fig. 1. Table 1: Main data of a face-milled generated spiral bevel gear

Blank data Parameter

Value

Parameter

Value

Gear tooth number Module Pinion handle Face width Outer addendum Face angle

33 4.8338 Right hand 27.5000 mm 1.7600 mm 76.1167°

Pinion tooth number Shaft angle Mean spiral angle Clearance Outer dedendum Root angle

9 90.0000° 32° 1.0300 mm 7.6700 mm 69.5833°

Blade data Parameter

Value

Parameter

Value

Average radius Pressure angle

63.5000 mm 22.0000°

Point width Fillet radius

2.5400 mm 1.5240 mm

Machine-settings Parameter

Value

Parameter

Value

Radial setting Sliding base Blank offset Roll ratio

64.3718 mm 0.0000 mm -0.2071 mm 1.0323

Cradle angle Machine center to back Machine root angle Modified roll coefficients

-56.7800° 0.0000 mm 69.5900° 0.0000

11

Contact point

Contact area

Major axis of contact ellipse

Contact line

(a) An example of contact line and contact area

(b) An idea type of contact line used for five-axis flank milling

Figure 6: Contact line and area.

227

3.4. Contact line and area

228

When a pair of spiral bevel or hypoid gears are meshing to transfer power, the contact line and area on the tooth

229

surface are formed during the meshing process, as shown in Fig. 6 (a). Once both tooth surfaces of a pair of gear drive

230

are obtained, the contact line and area can be calculated by TCA [3–5]. For a gear drive, edge contact should be avoided

231

as possibly as it can to improve work performances and serve life. The example in Fig. 6 (a) is an acceptable case that

232

edge contact is not happened. However, the contact line and area will be different due to manufacturing errors, load

233

and errors of alignment. A good design of tooth surface should be capable of avoiding the edge contact under those

234

changes. By considering those changes, an idea contact line rather than a general case is preferable in the initial design

235

of tooth surface. The idea contact line is usually chosen as the middle of a pair of tooth surfaces of a gear drive [3–5],

236

as shown in Fig. 7 (b). Generally, the idea contact line is comparative more robust to the aforementioned changes than

237

the contact line and area of a general case. Hence, this idea contact line is used in the further tool path planning of

238

five-axis flank milling.

239 240

241

The idea contact line can be defined according to the blank parameters, as shown in Fig. 7 (a). Mathematically, a point q on this contact line can be determined as    hq + c   · cos Γg  rq = Aq · sin Γg − bqg − 2   hq + c   · sin Γg .  zq = Aq · cos Γg + bqg − 2

(15)

242

where c and Γg are two given gear blank parameters, clearance and pitch angle, respectively; Aq , hq and bqg are the

243

pith cone distance, tooth depth and dedendum corresponding to point q, respectively. When Aq is given to sample the

244

contact line, hq and bqg can be derived from the gear blank parameters [46].

245

Once q is obtained with the gear blank parameters, the contact points on the tooth surface can also obtained as

12

Contact line

hq bqg

Pitch cone

og

hq+c 2 q xg Concave side of tooth surface

zg

rq qo Aq Γg og

qi q

zg

Plane ogxgzg

zq

(a) Define the contact line with gear blank parameters

Convex side of tooth surface

(b) Determine the contact point on the tooth surface

Figure 7: The definition of contact line with gear blank.

246

shown in Fig. 7 (b). When point q is rotated along with zg , a circular curve is formed and intersected with the booth

247

sides of tooth surface as qi and qo , respectively. qi and qo are the contact points on the convex and concave sides of

248

the tooth surface, respectively. To calculate a contact point on the tooth surface, we have a system of two equations in

249

two unknowns

250

 

Z(h∗ , φ∗c ) = zq

 X 2 (h∗ , φ∗ ) + Y 2 (h∗ , φ∗ ) = r2 . c c q

(16)

251

where X(h∗ , φ∗c ), Y (h∗ , φ∗c ), Z(h∗ , φ∗c ) are the coordinates of the tooth surface in Sg . Once both unknowns h∗ and φ∗c

252

are calculated according to Eq. (16), the contact line is obtained on the tooth surface.

253

At every contact point, the contact ellipse can be calculated according to the tooth surface geometry and the elastic

254

deformation at the contact point [3, 4]. Subsequently, the whole contact area is obtained by considering all contact

255

ellipses along the contact line. To make it simple, here we just consider the contact area as a small region around the

256

contact line. For the example given with the data in Tab. 1, the contact line are calculated as shown in Fig. 8 (a). It will

257

be used to plan the tool path of five-axis flank milling.

258

4. Five-axis flank milling of the tooth surface

259

With the models of tooth surface and contact line, the tool path planning for five-axis flank milling is implemented

260

with two steps. First, the tool path strategy based on the necessary conditions is used to make the cutter envelope

261

surface tangent with the designed surface along the contact line. Second, cutter orientations are optimized to obtain the

262

minimal geometric deviations of the contact area.

13

263

4.1. Tool path planning strategy based on the necessary conditions

264

As shown in Fig. 8, a conical cutter is used to machine the convex side of the tooth surface. According to the

265

necessary conditions stated in Subsection 2.2, ph should be the same point on the cutter axis l during the cutter moving

266

along the contact line. Subsequently, hq and ρq are constant. By considering these conditions, the tool path planning

267

strategy is shown in Fig. 8 (a). At a point q on the contact line, the cutter axis l and cutter tip point oc can be represented

268

in a local coordinate system Sq (q; nq , t, d). nq is the unit normal vector of both cutter envelope surface and designed

269

surface. t is the unit tangent vector of the contact line at q. d is obtained as d = nq × t. Since nq , t and d can be

270

calculated in coordinate system Sg , l and oc can also be obtained in Sg .

Conctact line Tooth Surface zg

d αc

e μ

q

pq

oc

xg

nq

ph

t nq

q

ρq

yg og

l

e

l

hq

oc (b)

(a)

Figure 8: Tool path planning strategy.

271 272

As shown in Fig. 8 (b), there is a plane passing through q and l, and e is the unit vector of the line intersected by the cutter surface and this plane. Then we have

273

l = cos αc · e + sin αc · nq

274

where αc is taper angle of the cutter. Since e is also a vector in the plane tqd, e can be defined with an angle parameter

275

µ, which is the angle from t to e. We have e = cos µ · t + sin µ · d.

(18)

l (φ) = cos αc · [cos (µ (φ)) · t (φ) + sin (µ (φ)) · d (φ)] + sin αc · nq (φ) .

(19)

276

277

By submitting Eq. (18) into Eq. (17), we have

278

279

(17)

Subsequently, oc can be calculated as oc (φ) = ph (φ) − hq · l (φ)

280

= q(φ) + ρq (φ) · nq (φ) − hq · l (φ) .

(20)

281

With Eqs. (19) and (20), the tool path is planned that the cutter envelope surface is tangent with the tooth surface along

282

the contact line. 14

283

Furthermore, hq and µ could be the variables used to find an optimal tool path. hq is a constant during the cutter

284

moving along the contact line, but µ could be varied. Here, a polynomial function in terms of the motion parameter φ

285

is applied to define the µ as

286

µ (φ) = u1 + u2 · φ + u3 · φ2 + u4 · φ3

287

where ui (i = 1 ∼ 4) are the coefficients of the polynomials. Assuming x = [u1 , u2 , u3 , u4 , hq ], the optimization

(21)

288

problem with respect to the variable x is proposed as follows to minimize the geometric deviations of the contact area.

289

4.2. Optimization model to minimize the geometric deviations of the contact area

290

Once the machined surface and designed surface are tangent along the designed line, both of them have the same

291

normal curvature at every point on the tangent direction of the contact line. Subsequently, an effective way to reduce

292

the geometric deviations around this point is to minimize the relative normal curvature of direction d [45]. Therefore,

293

the geometric deviations of the contact area can be reduced by minimizing the overall relative normal curvatures of direction d along the contact line. d 1/κd: radius of curvature of the designed surface on direction d q nq 1/κe: radius of curvature of the envelope surface on direction d

L

ε

Figure 9: A closed-form approximate model of geometric deviation. 294 295

Geometric deviation is usually calculated by complicated numerical methods [22–24]. While the machined surface

296

and designed surface are tangent along the designed line, the geometric deviation at a point on the contact area can be

297

simply calculated with an approximate model ε, as shown in Fig. 9. Two curves are obtained by intersecting the plane

298

nq qd with cutter envelope surface and designed surface. The normal curvatures of both cutter envelope surface and

299

designed surface along the direction d are κe and κd , respectively. Then the approximate model ε of direction d can

300

be calculated as a closed-form result ε=

301

302

L2 · (κd − κe ) 2

(22)

where L is the distance measured on direction d for the distance from q to the measure point; κd − κe is the relative

303

normal curvature of direction d. According to Eq. (22), the way to reduce ε of direction d is minimizing the relative

304

normal curvature.

305

By submitting Eqs. (19) and (20) into Eqs. (1) and (2), the cutter envelop surface can be obtained. Subsequently, the

306

normal curvature of the cutter envelop surface can be obtained as stated in Appendix. It is a function of x and written 15

307 308

as κe (x). For a set of data points qi (i = 1 ∼ N ) sampled from the contact line, a mathematical model is proposed to

minimize the least squares of the relative normal curvatures, and it is written as min

309

N X 1

(κd − κe (x))2

(23)

s.t. µmin ≤ µi ≤ µmax , i = 1 ∼ N 310

where x is optimal variable; µmin and µmax are used to define the range of µi . To make the topic more clear, the

311

constraints for other purposes, such as interference avoidance, are not discussed here. Eq. (23) is used to the least

312

squares of the relative normal curvatures. For some cases, the absolute value of the relative normal curvature is close

313

to 0, so it is better to use the radius of curvature to replace the normal curvature in Eq. (23). Subsequently, Eq. (23) is

314

replaced as N X 1 1 )2 min ( − κ κ d e (x) 1

315

(24)

s.t. µmin ≤ µi ≤ µmax , i = 1 ∼ N. 316

5. Example and discussion

317

The proposed approach is applied to the example of Tab. 1 stated in Section 3. The parameters of the conical cutter

318

are chosen with 15° taper angle, 1.5 mm bottom radius and 1.0 mm fillet radius. The range of hq is given as [2 mm

319 320

3.5 mm], and this has been checked without interference. The optimization model is established according to Eq. (24).   For this model, the range of µi is chosen as π6 , 5π 6 , and N = 21. According to the optimization result, the machined

321

surface corresponding to the optimal tool path is obtained and modeled in CATIA V5R20. Consequently, the geometric

322

deviations can be obtained by comparing the machined surfaces with the designed surface in CATIA V5R20.

323

As an illustration, several sampling points are chosen to show the result. As shown in Fig. 10, the sampling

324

points are chosen from three curves, L0 , L1 and L2 . L0 is the contact line. L1 and L2 are parallel to L0 with offset

325

distance 1 mm. The geometric deviations on these chosen points are shown in Tab. 2. Moreover, as a comparison, the

326

same cutter and procedure are applied for the previous approach [20] stated as Fig. 1 in Introduction. The results are illustrated as follows.

L1 L0 L2

p11 p01 p21

p12 p02 p22

p13 p03 p23

p14 p04 p24

p15 p05 p25

p16 p06 p26

p17 p07 p27

Figure 10: Sampling points on the designed surface. 327 328 329

(1) For the points on L0 , all the geometric deviations generated from the proposed approach are 0. This validates that the machined surface and designed surface are tangent along the designed line. 16

330

(2) The comparison shows the proposed approach reduces the geometric deviations of the contact line and area.

331

(3) While we check the geometric deviations on the contact area, they are less than 13 µm. If there is no other error

332

sources, the AGMA quality number of this example is 10, which is acceptable in this case. Table 2: Geometric deviations at chosen points (µm)

Curve Li i=0 i=1 i=2

Method

pi1

pi2

pi3

pi4

pi5

pi6

pi7

Proposed approach Previous approach Proposed approach Previous approach Proposed approach Previous approach

0 2.147 10.235 14.925 11.213 17.231

0 2.241 9.753 14.786 11.306 18.827

0 1.968 10.212 15.320 10.898 16.762

0 2.358 10.304 16.724 12.021 16.676

0 2.017 11.103 15.035 12.534 18.139

0 2.342 10.967 15.901 12.305 18.573

0 2.110 10.892 15.216 12.430 17.971

333

The example focuses on the contact area of the tooth surface. If we consider the whole tooth surface, there are

334

two choices. One is applying multiple passes to machine whole tooth surface. Multiple passes have been used to

335

end milling (both 3-axis and multi-axis) the tooth surface [17–19]. It is also can be applied in the proposed approach,

336

although the quality and efficiency would be inferior to one pass. Another choice is using a cylindrical cutter to cut each

337

side of tooth surface with one pass. As stated in Subsection 2.2, hq can be varied to satisfy the necessary conditions

338

while the cutter is a cylinder. The determination of hq can be referred to [20]. The same procedure as the proposed

339

example can be applied to the cylindrical cutter with two modifications. First, cutter shape parameters are changed.

340

Second, hq is not an optimization variable.

341

6. Conclusions

342

This research has proposed an innovative method to determine cutter locations and orientations for five-axis flank

343

milling of the spiral bevel and hypoid gears with high accuracy. The main features of this approach are to identify the

344

critical contact area of the tooth surfaces according to the gear meshing characteristics and plan tool paths to cut the

345

contact area with high accuracy. The academic contributions of this approach include (1) formulation of the necessary

346

conditions of the cutter envelope surface tangent with the designed surface along a designed line in five-axis milling,

347

(2) an closed-form approximate model is used to simplify the calculation of the geometric deviations, (3) establishment

348

of the optimization model of tool orientation to minimize the geometric deviation of the contact area and the cutter

349

envelope surface. This approach can significantly promote efficient and accurate flank milling of these gears in industry.

350

It is still a new topic for five-axis flank milling spiral bevel and hypoid gears. The future work could be conducted

351

to the following two aspects. (1) Some technologies of five-axis flank milling, such as interference check, machining

352

error control and machining efficiency improvement, can be applied to cut spiral bevel and hypoid gears. (2) The

353

designed tooth surface used in this paper is a given model calculated according to conventional manufacturing meth-

354

ods. However, from a design view, the tooth surface model could be changed to obtain better work performances.

355

Subsequently, the optimized model of tool path of five-axis flank milling need be obtained to achieve the optimal work

356

performances.

17

357

Acknowledgment

358

Financial supports from National Natural Science Foundation of China (No. 51275530 and No. 51535012) to J.

359

Tang, as well as National Natural Science Foundation of China (No. 61572527) and Hunan Province Science and

360

Technology Project (No. 2014FJ2008) to Shengjun Liu, are gratefully acknowledged. The authors would like to thank

361

Dr. Limin Zhu for the valuable discussions with him.

362

A. Appendix

363

A.1. The calculations of envelope surface and its normal curvature

364 365

In order to calculate the normal curvatures, the partial derivatives of r(h, φ) and n(h, φ) should be calculated first. According to Eq. (2), we have f1 · n = f2 · l + f3 · vh + f4 · (l × vh )

366

367

where

368

369

370 371

372

(A.1)

q 2 2 f1 = (l × vh ) , f2 = cos α · vh2 , f3 = − cos α · (l · vh ) , f4 = ± (l × vh ) − cos2 α · vh2

Then partial derivatives of n are calculated as  1 ∂f1 ·n+ nh = · − f1 ∂h  1 ∂f1 nφ = · − ·n+ f1 ∂φ where

  ∂f2 ∂f3 ∂vh ∂f4 ∂vh ·l+ · vh + f3 · + · (l × vh ) + f4 · l × ∂h ∂h ∂h ∂h ∂h   dl ∂f2 dl ∂f3 ∂vh ∂f4 ∂vh · l + f2 · + · vh + f3 · + · (l × vh ) + f4 · × vh + l × ∂φ dφ ∂φ ∂φ ∂φ dφ ∂φ (A.2) ∂f1 ∂h ∂f2 ∂h ∂f3 ∂h ∂f4 ∂h ∂f1 ∂φ ∂f2 ∂φ ∂f3 ∂φ ∂f4 ∂φ ∂vh ∂h ∂vh ∂φ

  ∂ (l × vh ) ∂vh = 2 · (l × vh ) · l × ∂h ∂h dα 2 ∂vh = − sin α · · v + 2 · cos α · vh · dh h ∂h dα = sin α · · (l · vh ) dh   1 ∂f1 dα 2 ∂vh = · + sin 2α · · vh − 2 · cos2 α · vh · 2 · f4 ∂h dh ∂h   dl ∂vh = 2 · (l × vh ) · × vh + l × dφ ∂φ ∂vh = 2 · cos α · vh · ∂φ d (l · vh ) = − cos α · dφ   1 ∂f1 ∂vh 2 = · − 2 · cos α · vh · 2 · f4 ∂φ ∂φ dl(φ) = dφ 2 d oc (φ) d2 l(φ) = + h · . dφ2 dφ2 = 2 · (l × vh ) ·

18

373

Furthermore, the derivatives of the r(h, φ) can be calculated according to Eq. (1) as ∂r(h, φ) dρ(h) = l(φ) + · n(h, φ) + ρ(h) · nh (h, φ) ∂h dh ∂r(h, φ) doc (φ) dl(φ) rφ = = +h· + ρ(h) · nφ (h, φ) ∂φ dφ dφ rh =

374

(A.3)

= vh (h, φ) + ρ(h) · nφ (h, φ). According to the knowledge of differential geometry, the normal curvature of the envelope surface along a direction

375 376

can be calculated if the direction is given. Generally, the direction vector is expressed as

377

dr = rh · dh + rφ · dφ.

378

One parameter is used to define the direction vector with either u = dh/dφ (dφ 6= 0) or v = dφ/dh (dh 6= 0). Taking

379

(A.4)

the example of parameter u, the normal curvature along dr is determined by the equation [3, 47] κ=

380

L · u2 + 2 · M · u + N II −dr · dn = = I dr2 E · u2 + 2 · F · u + G

(A.5)

381

where I, II are first and second fundamental forms, respectively. E, F and G are the coefficients of the first funda-

382

mental form, and L, M and N are the coefficients of the second fundamental form. These coefficients are obtained

383

as

384

E = r2h , F = rh · rφ , G = r2φ

(A.6)

385 386

L = −rh · nh , M = −rh · nφ = −rφ · nh , N = −rφ · nφ

(A.7)

387 388

Since all the above calculations are conducted as the closed-form results, the result of the normal curvature of the envelope surface is also a closed-form result.

Reference [1] H. T¨onshoff, C. Gey, N. Rackow, Flank milling optimization-the flamingo project, Air & Space Europe 3 (3) (2001) 60–63. [2] H.-T. Young, L.-C. Chuang, K. Gerschwiler, S. Kamps, A five-axis rough machining approach for a centrifugal impeller, The International Journal of Advanced Manufacturing Technology 23 (3-4) (2004) 233–239. [3] F. L. Litvin, A. Fuentes, Gear geometry and applied theory, Cambridge University Press, 2004. [4] F. L. Litvin, Theory of gearing, AVSCOM technical report, National Aeronautics and Space Administration, Scientific and Technical Information Division, 1989. [5] F. L. Litvin, Y. Zhang, Local synthesis and tooth contact analysis of face-milled spiral bevel gears, Tech. rep., DTIC Document (1991). [6] C. B. Tsay, J. Y. Lin, A mathematical model for the tooth geometry of hypoid gears, Mathematical and Computer Modelling 18 (2) (1993) 23 – 34. 19

[7] Y. Zhou, Z. C. Chen, A new geometric meshing theory for a closed-form vector representation of the face-milled generated gear tooth surface and its curvature analysis, Mechanism and Machine Theory 83 (2015) 91–108. [8] Y. P. Shih, G. C. Lin, Z. H. Fong, Mathematical model for a universal face hobbing hypoid gear generator, Journal of Mechanical Design 129 (1) (2007) 38–47. [9] M. Vimercati, Mathematical model for tooth surfaces representation of face-hobbed hypoid gears and its application to contact analysis and stress calculation, Mechanism and Machine Theory 42 (6) (2007) 668–690. [10] M. Lelkes, D. Play, J. Marialigeti, Numerical determination of cutting parameters for the control of Klingelnberg spiral bevel gear geometry, Journal of Mechanical Design 124 (4) (2002) 761–771. [11] J. Achtmann, G. B¨ar, Optimized bearing ellipses of hypoid gears, Journal of Mechanical Design 125 (4) (2003) 739–745. [12] Q. Fan, Computerized modeling and simulation of spiral bevel and hypoid gears manufactured by Gleason face hobbing process, Journal of Mechanical Design 128 (6) (2006) 1315–1327. [13] Q. Fan, Enhanced algorithms of contact simulation for hypoid gear drives produced by face-milling and facehobbing processes, Journal of Mechanical Design 129 (1) (2007) 31–37. [14] V. V. Simon, Design and manufacture of spiral bevel gears with reduced transmission errors, Journal of Mechanical Design 131 (4) (2009) 041007. [15] V. V. Simon, Advanced manufacture of spiral bevel gears on CNC hypoid generating machine, Journal of Mechanical Design 132 (3) (2010) 031001. [16] V. V. Simon, Generation of hypoid gears on CNC hypoid generator, Journal of Mechanical Design 133 (12) (2011) 121003. [17] S. Suh, W. Jih, H. Hong, D. Chung, Sculptured surface machining of spiral bevel gears with CNC milling, International Journal of Machine Tools and Manufacture 41 (6) (2001) 833–850. [18] S.-H. Suh, D.-H. Jung, S.-W. Lee, E.-S. Lee, Modelling, implementation, and manufacturing of spiral bevel gears with crown, The International Journal of Advanced Manufacturing Technology 21 (10-11) (2003) 775–786. [19] J. T. Alves, M. Guingand, J.-P. de Vaujany, Designing and manufacturing spiral bevel gears using 5-axis computer numerical control (cnc) milling machines, Journal of Mechanical Design 135 (2) (2013) 024502. [20] Z. Yuansheng, Five-axis flank milling and modeling the spiral bevel gear with a ruled tooth surface design, Ph.D. thesis, Concordia University (2015). [21] C. Lartigue, E. Duc, A. Affouard, Tool path deformation in 5-axis flank milling using envelope surface, ComputerAided Design 35 (4) (2003) 375–382. [22] L. Zhu, X. Zhang, G. Zheng, H. Ding, Analytical expression of the swept surface of a rotary cutter using the envelope theory of sphere congruence, Journal of Manufacturing Science and Engineering 131 (4) (2009) 041017. [23] P.-Y. Pechard, C. Tournier, C. Lartigue, J.-P. Lugarini, Geometrical deviations versus smoothness in 5-axis highspeed flank milling, International Journal of Machine Tools and Manufacture 49 (6) (2009) 454–461. [24] Y. Zhou, Z. C. Chen, X. Yang, An accurate, efficient envelope approach to modeling the geometric deviation of the machined surface for a specific five-axis CNC machine tool, International Journal of Machine Tools and Manufacture. [25] S. Bedi, S. Mann, C. Menzel, Flank milling with flat end milling cutters, Computer-Aided Design 35 (3) (2003) 293–300. [26] C. Menzel, S. Bedi, S. Mann, Triple tangent flank milling of ruled surfaces, Computer-Aided Design 36 (3) (2004) 289–296.

20

[27] C. Li, S. Mann, S. Bedi, Error measurements for flank milling, Computer-Aided Design 37 (14) (2005) 1459– 1468. [28] J. C. Chiou, Accurate tool position for five-axis ruled surface machining by swept envelope approach, ComputerAided Design 36 (10) (2004) 967–974. [29] J. Senatore, F. Monies, J.-M. Redonnet, W. Rubio, Improved positioning for side milling of ruled surfaces: analysis of the rotation axis’s influence on machining error, International Journal of Machine Tools and Manufacture 47 (6) (2007) 934–945. [30] C.-H. Chu, J.-T. Chen, Tool path planning for five-axis flank milling with developable surface approximation, The International Journal of Advanced Manufacturing Technology 29 (7-8) (2006) 707–713. [31] C. Chu, W. Huang, Y. Hsu, Machining accuracy improvement in five-axis flank milling of ruled surfaces, International Journal of Machine Tools and Manufacture 48 (7) (2008) 914–921. [32] P.-H. Wu, Y.-W. Li, C.-H. Chu, Optimized tool path generation based on dynamic programming for five-axis flank milling of rule surface, International Journal of Machine Tools and Manufacture 48 (11) (2008) 1224–1233. [33] H.-T. Hsieh, C.-H. Chu, Particle swarm optimisation (PSO)-based tool path planning for 5-axis flank milling accelerated by graphics processing unit (GPU), International Journal of Computer Integrated Manufacturing 24 (7) (2011) 676–687. [34] J. Chaves-Jacob, G. Poulachon, E. Duc, New approach to 5-axis flank milling of free-form surfaces: Computation of adapted tool shape, Computer-Aided Design 41 (12) (2009) 918–929. [35] G. Zheng, L. Zhu, Q. Bi, Cutter size optimisation and interference-free tool path generation for five-axis flank milling of centrifugal impellers, International Journal of Production Research 50 (23) (2012) 6667–6678. [36] L. Zhu, H. Ding, Y. Xiong, Simultaneous optimization of tool path and shape for five-axis flank milling, Computer-Aided Design 44 (12) (2012) 1229–1234. [37] Z. Li, L. Zhu, Envelope surface modeling and tool path optimization for five-axis flank milling considering cutter runout, Journal of Manufacturing Science and Engineering 136 (4) (2014) 041021. [38] L. Zhu, Y. Lu, Geometric conditions for tangent continuity of swept tool envelopes with application to multi-pass flank milling, Computer-Aided Design 59 (2015) 43–49. [39] L. Zhu, G. Zheng, H. Ding, Y. Xiong, Global optimization of tool path for five-axis flank milling with a conical cutter, Computer-Aided Design 42 (10) (2010) 903–910. [40] H. Gong, L.-X. Cao, J. Liu, Improved positioning of cylindrical cutter for flank milling ruled surfaces, ComputerAided Design 37 (12) (2005) 1205–1213. [41] H. Gong, N. Wang, Optimize tool paths of flank milling with generic cutters based on approximation using the tool envelope surface, Computer-Aided Design 41 (12) (2009) 981–989. [42] H. Gong, N. Wang, 5-axis flank milling free-form surfaces considering constraints, Computer-Aided Design 43 (6) (2011) 563–572. [43] R. F. Harik, H. Gong, A. Bernard, 5-axis flank milling: A state-of-the-art review, Computer-Aided Design 45 (3) (2013) 796–808. [44] W. Wang, K. Wang, Geometric modeling for swept volume of moving solids, IEEE Computer Graphics and Applications 6 (12) (1986) 8–17. [45] H. Gong, L.-X. Cao, J. Liu, Second order approximation of tool envelope surface for 5-axis machining with single point contact, Computer-Aided Design 40 (5) (2008) 604–615. [46] A. Standard, Design manual for bevel gears, ANSI/AGMA (2005) 2005–D03. [47] A. N. Pressley, Elementary differential geometry, Springer, 2010. 21

Figure

Concave side of tooth suface

Convex side of tooth suface Cutter

Toe P hq

q l

Heel oc

Contact line

Tooth surface

(b) A tool path planning strategy

(a) Contact line on the tooth surface

Fig. 1 Five-axis flank milling the tooth surfaces with an existing tool path planning strategy

θ

n p

α

ph

ρ l

Generatrix

h

oc

Fig. 2 A general surface of revolution

Designed line q(ϕ) CC line p(ϕ) ph

ρq nq q

p

l

hq

oc

Fig. 3 Formulations of the necessary conditions

yg (1)

Cradle Head-cutter

om,oc ym

xg ym xc q ϕ c yc om,oc xm og

zm,zc

Blades

(2)

xm

om

on

zg

zm

xb ϕ b γm ϕb

(3) ob

oa

zb

yb

Gear blank

Fig. 4 Generation motions and kinematics of face-milled generated gears

concave side ph θ

convex side n straight line Xw

αg

h circular fillet og

xg Ru

yg

φ

n

rf Pw

zg, lg

Fig. 5 Blade cutting edges

Contact point

Contact area

Major axis of contact ellipse

Contact line

(a) An example of contact line and contact area

(b) An idea type of contact line used for five-axis flank milling

Fig. 6 Contact line and area Contact line

hq bqg

Pitch cone

og

hq+c 2 q xg Concave side of tooth surface

zg

rq qo Aq Γg og

qi q

zg

Convex side of tooth surface

Plane ogxgzg zq

(a) Define the contact line with gear blank parameters

(b) Determine the contact point on the tooth surface

Fig. 7 The definition of contact line with gear blank

Conctact line

zg

Tooth Surface

d

q

e μ pq

αc e

l

xg

oc

(a)

nq

ph

t nq

q

yg og

l

(b)

Fig. 8 Tool path planning strategy

ρq oc

hq

d Intersection curve of the designed surface κd q nq

κe Intersection curve of the envelope surface

L

ε

Fig. 9 A closed-form approximate model of geometric deviation

L1 L0 L2

p11 p01 p21

p12 p02 p22

p13 p03 p23

p14 p04 p24

p15 p05 p25

p16 p06 p26

p17 p07 p27

Fig. 10 Sampling points on the designed surface