Design of high-reduction hypoid gears meshing in plane of ... · From this viewpoint, the design of the tooth surface can be independent from the tooth cutting process. Thus, the

Bulletin of the JSME

Journal of Advanced Mechanical Design, Systems, and ManufacturingVol.11, No.6, 2017

Paper No.17-00317© 2017 The Japan Society of Mechanical Engineers[DOI: 10.1299/jamdsm.2017jamdsm0068]

Design of high-reduction hypoid gears meshing in plane of action

Atsushi SUZUKI *, Ichiro TARUTANI* and Takayuki AOYAMA* * Toyota Central R&D Labs., Inc.

41-1, Yokomichi, Nagakute, Aichi, 480-1192, Japan E-mail: [email protected]

1. Introduction

Some orthogonal layouts of gear pairs can perform various functions, such as acting as the final gear drive of an

automobile (Miyamura et.al, 2013), the reduction gear of an electric motor (Ohshima and Yoshino, 2006), or a spinning reel (Inoue and Kurokawa, 2012). Hypoid gears and face gears (Litvin et.al, 2005) (Tsuji et.al, 2009) are commonly selected to transmit loads and change the direction of rotation. Moreover, higher reduction ratio gears than the final gear drives are required for the use of high-speed electric motors (Aiki et.al, 2017). The commonly used geometries of hypoid and bevel gears are dependent on the type of tooth cutting process, such as face milling and face hobbing (Stadtfeld, 2014). These manufacturing methods enable mass production in the automotive industry. In contrast, some face gears are manufactured using general-purpose machining centers because of their unique tooth surfaces (Kawasaki et.al, 2011). Some spiral bevel gears have also been manufactured using five-axis computer numerical control (CNC) milling machines (Alves et.al, 2013). Recent technical progress in machining center design has enabled the replacement of conventional gears by producing a new type of tooth surface.

From this viewpoint, the design of the tooth surface can be independent from the tooth cutting process. Thus, the authors refer to a previously developed design theory (Honda, 2009, 2016a, 2016b, 2016c) to obtain hypoid gears with a higher performance than face gears (Honda, 2016d) (Aiki et.al, 2017). The theory can define the surfaces of action without the design limitations of manufacturing methods. Then, coordinate transformations can be performed to obtain the conjugate tooth surfaces. The characteristics of gear meshing can be controlled by defining the surface of action. This design theory has been applied to the production of final gear drives (Miyamura et.al, 2013).

Another issue in the design of hypoid and face gears is obtaining an appropriate tooth bearing when misalignment has occurred. Misalignment may induce the edge contact of tooth surfaces, affecting the vibration and strength of the gears. Some studies on face gears have avoided edge contact by introducing tooth surface modifications (Inoue and Kurokawa, 2012).

This paper describes a design method involving the meshing of a high-reduction hypoid gear in the plane of action. The plane of action of a hypoid gear may be selected such that it has features similar to those of cylindrical involute gears as shown in Table 1. The cylindrical involute gears are unaffected by translational assembly error. The design

1

Received: 26 June 2017; Revised: 10 August 2017; Accepted: 20 September 2017

Abstract This paper proposes a method of directly designing the surface of action of a hypoid gear. In the proposed method, the characteristics of the gear meshing are independent of the manufacturing process. The conjugate gear surfaces can then be accurately obtained by coordinate transformation. A plane was selected as the surface of action to achieve hypoid gears with a higher performance than face gears. The proposed hypoid gear may have the same features as cylindrical involute gears that also mesh in the plane of action and are unaffected by translational assembly error. A design example is presented in this paper to verify that tooth surface meshing in the plane of action can be achieved for a high-reduction ratio hypoid gear. The influence of different types of assembly error on the tooth flank error was examined numerically by comparing with face gears.

Keywords : Hypoid gear, Surface of action, Contact ratio, Misalignment, Assembly error, Face gear

2© 2017 The Japan Society of Mechanical Engineers[DOI: 10.1299/jamdsm.2017jamdsm0068]

Suzuki, Tarutani and Aoyama,Journal of Advanced Mechanical Design, Systems, and Manufacturing, Vol.11, No.6 (2017)

examples presented in this paper demonstrate that such hypoid gear can be designed and reveal some advantages of these gears in comparison with face gears with regard to the tooth flank error caused by assembly error.

Table 1 Classification of targeted gears

Gear type Parameters

Cylindrical involute gear

Face gear Conventional hypoid gear

Proposed hypoid gear

Shaft angle Σ [deg] 0 90 90 90

Tooth type of pinion Involute Involute Surface defined by

tooth cutting Surface defined by

plane of action

Tooth type of gear Involute Conjugate surface of involute pinion

Surface defined by tooth cutting

Surface defined by plane of action

External shape of pinion teeth

Cylinder Cylinder Cone Cone

External shape of gear teeth

Cylinder Flat Cone Cone

Shape of surface of action

Plane Curved surface Curved surface Plane

Sensitivity to misalignment

Low (translational directions)

Low (pinion-axis direction)

High (all directions)

Low (translational directions)

2. Design procedures 2.1 Definition of planar surface of action

The design flow for the proposed hypoid gear that meshes in the plane of action is shown in Fig. 1 (Honda, 2016b).

The locations of the two rotational axes of the pinion and gear, the shaft angle Σ, and the pinion offset E can be used to obtain the coordinate systems Cs (uc, vc, zc), C1 (u1c, v1c, z1c), and C2 (u2c, v2c, z2c) as shown in Fig. 2 (Honda, 2016a).

Fig. 1 Design procedure of proposed hypoid gear.

Coordinate systems Cs, C1, C

2

Design point P0 (uc0

, vc0, zc0

)

Paths of contact g0D, g

0C

Surfaces of action SwD, SwC

Conjugate tooth surfaces

Basic dimensions

Boundaries of tooth

Locations of two axes Σ, E Gear ratio i

0 (= N

2/N

1)

2

.

The axes z1c and z2c of the coordinate system C1 and C2 are defined as the rotational axes of the pinion and the gear

as shown in Fig. 2. These rotational axes are in the same directions of the angular velocity of the pinion 𝜔𝜔1��⃗ and that of the gear 𝜔𝜔2��⃗ . The coordinate systems C1 and C2 have the common vertical axes v1c and v2c which are defined as the direction of the exterior product 𝜔𝜔2��⃗ × 𝜔𝜔1��⃗ . The axes u1c and u2c are set perpendicular to the axes z1c and v1c, and the



axes z2c and v2c, respectively. The coordinate system CS has the common vertical axis vc with v1c and v2c. The axis zc of the coordinate system CS is in the direction of relative angular velocity 𝜔𝜔𝑟𝑟��⃗ = 𝜔𝜔1��⃗ − 𝜔𝜔2��⃗ as shown in Fig. 2(a). The axis uc is set perpendicular to the axis zc and vc.

The design point P0 (uc0, vc0, zc0), one of the contact points on the coordinate system Cs, is defined as

( ) ( ) ( )

( )( )

00

0

2 220 2 0

0

tan sin sinsin( )

0

cossin

s sc

c

cs c sc

s

Eu

v

R v uz

ϕ Σ − Γ Γ=

Σ=

− + Γ=

Γ

(1)

where

( ) ( )( )

( ) ( )

0

2

1 2

sin sin

tantan tan

s s

scs

s s

cs cs

i

Ev

v v E

Γ = Σ − Γ

Γ=

Σ − Γ + Γ

= −

ϕ0 is the inclination angle of the path of contact g0, R20 is the radius of the design point P0, and Γs is the angle between the axes zc and z2c. The angle Γs shown in Fig. 2(a) corresponds to the direction of the relative angular velocity of 𝜔𝜔𝑟𝑟��⃗ = 𝜔𝜔1��⃗ − 𝜔𝜔2��⃗ . The pinion offset E, the length vcs1, vcs2 from the origin of Cs at the origin of C1 and C2 are defined in Fig. 2(b). The overview of three coordinate systems is illustrated in Fig. 2(c).

Fig. 2 Coordinate systems applied for the design.

The paths of contact g0D and g0C of the drive and coast sides are located to pass through P0 in the direction of the

normal, which are defined in the coordinate system Cs by the pair of inclination angles (φn0D, ϕ0) for drive side and (φn0C, ϕ0) for coast side (Honda, 2016b). The inclination angles mean the pressure angles φn0D, φn0C and the helix angle ϕ0 in the general design theory. The directions of the paths of contact are described in Fig. 3, where vc = 0, uc = uc0, and zc = zc0 are defined as the plane SH, Sp, and SS. The plane Sn is defined including paths of contact g0D and g0C with

Gear

Pinion

u2c

v1c, v2c, vc

z2c

z1c

u1c

E

zc uc Σ

Γs

The origin of Cs The origin of C2

The origin of C1

vcs2

vcs1

P0

ω2

ω1

𝜔𝜔2��⃗

𝜔𝜔1��⃗

𝜔𝜔𝑟𝑟��⃗

z1c

z2c Σ

zc

Γs

𝜔𝜔1��⃗

𝜔𝜔2��⃗

𝜔𝜔𝑟𝑟��⃗

(a) View from the axis vc

(b) View from the axis z2c (c) Overview of the coordinate systems

v1c, v2c,vc

u2c

zc' (projection of zc)

z1c

The origin of Cs

The origin of C2

The origin of C1

E vcs2

vcs1

3



inclination angle ϕ0 from the plane Ss. The inclination angle φn0D and φn0C are the angle from the intersection of the plane Sn and Sp to the paths of contact g0D and g0C on the plane Sn.

Fig. 3 Definitions of the directions for the paths of contact g0D and g0C.

( ) ( ) ( ) ( )( ) ( ) ( ) ( )( ) ( ) ( ) ( )( ) ( ) ( )

1 10 1 10 0 0 0

1 1 0 0 0

1 1 0 0 0

1 10 1 10 0 0

, cos cos tan

, , cos sin

, , sin cos

, cos sin

= + −

= +

= −

= + +

c Cq b b SW c Cq SW

c Cq c Cq s bc s

c Cq c Cq s bc s

c Cq b b SW Cq c

q z R q z

u z q z R

v z q z R

z z R z z

q q ϕ ϕ ϕ

q q c c

q q c c

q q ϕ ϕ

(2)

where q1 is the rotational angle of the pinion; zCq is the additional parameter to express the plane; Rb10 is the base circle radius of the pinion; and ϕb10 is the normal direction of the pinion. Equation (2) for the gear is derived by replacing parameters q1 ,Rb10 and ϕb10 with the rotational angle of the gear q2 , the base circle radius of the gear Rb20 and the normal direction of the gear ϕb20. Note that Eq. (2) represents a line of contact when q1 or q2 is constant, and the path of contact g0D or g0C when zCq = 0.

A plane of action can be expressed in the coordinate systems C1 (u1c, v1c, z1c) and C2 (u2c, v2c, z2c) as relations between Cs, as given by

( ) ( ) ( ) ( ) ( )( ) ( )( ) ( ) ( ) ( ) ( )

1 1 1 1

1 1 1 1

1 1 1 1

, , cos , sin

, ,

, , sin , cos

= Σ − Γ + Σ − Γ

= −

= − Σ − Γ + Σ − Γ

c Cq c Cq s c Cq s

c Cq c Cq cs

c Cq c Cq s c Cq s

u z u z z z

v z v z v

z z u z z z

q q q

q q

q q q

(3)

v1c, v2c, vc

zc

uc

z2c

z1c

Plane SH vc = 0

Plane Ss zc = zc0

Plane Sp uc = uc0

φn0D

φ0

Plane Sn P0(uc0, 0, zc0)

φn0C

g0D

g0C

Intersection of Sn and Sp

4

Planes of action SwD and SwC are defined as the planes created by g0D, g0C and the pitch line element (Honda,

2016b). The planes of action SwD and SwC are expressed in the coordinate system Cs (uc, vc, zc) with parameters q1 and zCq as



( ) ( ) ( ) ( ) ( )( ) ( )( ) ( ) ( ) ( ) ( )

2 2 2 2

2 2 2 2

2 2 2 2

, , cos , sin

, ,

, , sin , cos

= − Γ + Γ

= −

= − Γ − Γ

c Cq c Cq s c Cq s

c Cq c Cq cs

c Cq c Cq s c Cq s

u z u z z z

v z v z v

z z u z z z

q q q

q q

q q q

(4)

where some parameters in the Eqs. (2) – (4) are defined in the following equations:

( ) ( )( ) ( )

( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )

10 1 0 10 1 0 10

1 0 0 0

1 0 0 1

10 102

110 0 0 0

110 0 0 0

sin cos

cos sin

tan tan cos cos tan sin

sin sin sin cos sin cos

b p p

p c s c s

p c cs

n s s

b n s n s

R u v

u u zv v v

p

c c

c φ

φ φ ϕ ϕ

ϕ φ φ ϕ

−

−

= −

= Σ − Γ + Σ − Γ

= −

= −

= − Σ − Γ + Σ − Γ = Σ − Γ + Σ − Γ

(5)

( ) ( )( ) ( )

( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )

20 2 0 20 2 0 20

2 0 0 0

2 0 0 2

20 202

120 0 0 0

120 0 0 0

sin cos

cos sin

tan tan cos cos tan sin

sin sin sin cos sin cos

b p p

p c s c s

p c cs

n s s

b n s n s

R u v

u u zv v v

p

c c

c φ

φ φ ϕ ϕ

ϕ φ φ ϕ

−

−

= −

= − Γ + Γ

= −

= −

= Γ + Γ = Γ − Γ

(6)

( ) ( )( ) ( )

( ) ( )( ) ( )

0 0 0 0 0

0 0 0 0 0

0 02

10 0 0

10 0 0

cos sin

sin cos

tan tan cos

sin cos sin

c c s c s

bc c s c s

s s

s n

SW n

q u v

R u vp

c c

c c

c φ

φ φ ϕ

ϕ φ ϕ

−

−

= +

= −

= −

= − =

. (7)

Some more detailed expression of the Eqs. (2) – (7) are described in the published article (Honda, 2016b). 2.2 Conjugate tooth surfaces

The tooth surfaces of the hypoid gears can be determined by transforming their planes of action into the rotating

coordinate systems fixed to each gear (Honda, 2016b). The transformation into the coordinate system Cr1 (ur1c, vr1c, zr1c) yields the conjugate tooth surface of the pinion as

( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )( ) ( )

1 1 1 1 10 1 1 10

1 1 1 1 10 1 1 10

1 10 1

1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1

1 1 1 1

, , cos , sin

, , sin , cos

, , cos , sin

, , sin , cos

, ,

= +

= −

= −

= +

= −

=

c Cq c Cq c Cq

b Cq c Cq c Cq

r

r c Cq c Cq r b Cq r


r c Cq c Cq

q z u z v z

R z u z v z

u z q z R z

v z q z R z

z z z z

q q c q c

q q c q c

c c q

q q c q c

q q c q c

q q

. (8)

The coordinate system Cr1 rotates with the pinion and has the same origin as C1. In the same manner, the rotating coordinate system Cr2 (ur2c, vr2c, zr2c), which is fixed to the gear, yields the conjugate tooth surface of the gear as

5



( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )( ) ( )

2 2 2 2 20 2 2 20

2 2 2 2 20 2 2 20

2 20 2

2 2 2 2 2 2 2 2

2 2 2 2 2 2 2 2

2 2 2 2

, , cos , sin

, , sin , cos

, , cos , sin

, , sin , cos

, ,

= +

= −

= −

= +

= −

=

c Cq c Cq c Cq

b Cq c Cq c Cq

r



r c Cq c Cq

q z u z v z

R z u z v z

u z q z R z

v z q z R z

z z z z

q q c q c

q q c q c

c c q

q q c q c

q q c q c

q q

. (9)

2.3 Boundaries of tooth surface

The equivalent rack shown in Fig. 4 shows the definition of the profile of the tooth (Honda, 2016b). The rack is

defined on the plane Sn in Fig. 3 with the paths of contact g0D and g0C. The profiles of the pinion and gear are given by

( )

( )( )

( ) ( )( ) ( )( ) ( ) ( )

( ) ( )

20 200

2

0 0

0 0

10 2 20 0

0 0

1 2

2 cos

cossin

costan

cos tan sin sin sin

22

tan tan

b D b Dg D

g D nt n Ccr

n D n C

snt

cs s

cnk cr r

n D nt nt n C

d k d

Rp

Np

h

v R

th h c

A h A

p ϕ

φ φφ φ

φϕ ϕ

φ φ φ φ

−

=

−=

−

Γ=

− − + Γ

= − −− + −

= −

(10)

Fig. 4 Equivalent rack used to define the tooth profile.

Fig. 5 Tip and root cones to define the tooth width.

Pinion

Gear φn0C φn0D

hk Ad2

Ad1

cr

tcn

hcr

Tip

Root

z2c

z1c

Pitch cone

R1h z1t

z1h

γpw = π/2 – Γs Γgw = Γs

R2t R2h

Tip cone

Root cone

6

where tcn is the top land; cr is the clearance; and Ad2 is the gear addendum, which are given as input parameters. The subscripts D with pg0, Rb20 and ϕb20 indicate the dimensions for drive side which are obtained by Eq. (6) with a substitution of φn0 = φn0D.

The pitch cone angle Γgw of the gear is equal to Γs, and that of the pinion is γpw = p/2 – Γs as shown in Fig. 5 (Honda, 2016b). The tip and root cones can be selected to optimize each top land of the pinion and gear. The cone angles of the tip and root are selected to be equal to the pitch cone angle as shown in Fig. 5; that is, a parallel-depth tooth system is employed.



3. Estimation of tooth flank error by misalignment The types of misalignment illustrated in Fig. 6 can be converted to tooth flank error δLm as

2 2

2 2

2 2 2

sin

cos

∆ = ∆ Σ − ∆Σ∆ = ∆ − ∆∆ = ∆ − ∆ Σ + ∆Σ + ∆

u P zv E zz G P u v

ss

(11)

2 20 20 2 20 20 2 20cos cos cos sin sin= ∆ + ∆ + ∆b b bLm u v zδ ϕ c ϕ c ϕ (12)

where ∆E, ∆P, and ∆G are misalignments caused by translational assembly errors in the direction of the pinion offset, pinion axis, and gear axis, respectively; ∆Σ and ∆s are misalignments caused by rotational assembly errors about the v2- and u2-axes, respectively; and ϕb20 and c20 are the normal directions of the gear tooth expressed by Eq. (6) in the coordinate system C2 (Honda, 2016c). Equation (12) transforms each movement of the contact point into the tooth flank error in the direction of the tooth normal. 4. Design example

Table 2 gives the input parameters of the proposed hypoid gear. The proposed hypoid gear is compared with a face gear developed in a previous study (Honda, 2016d) (Aiki et.al, 2017). The gear dimensions of face gear are used in common with Table 2 except that Γgw= 90.0, γpw= 0, 2R1h= 17.2, and tooth depth of both gears = 2.1.

Table 2 Gear dimensions of the hypoid gear. Subscripts – D: drive side, C: coast side, 1: pinion, 2: gear

Fig. 6 Definition of misalignment caused by errors of supporting gears.

4.1 Planar surface of action Figure 7 shows the planes of action SwD and SwC. The above procedure can be used to consecutively determine the

design point P0, the paths of contact g0D and g0C, and the planes of action. Both sides of the planes of action intersect at the pitch line element, on which all contact points have a constant relative velocity. The other paths of contact that do not pass through P0 are also on the planes of action. All of the contact points satisfy the requirement for contact. Namely, the normal is perpendicular to relative velocity.

Input Units Pinion Gear Axis angle Σ deg 90.0 Gear ratio i0 - 15.1 Number of teeth N1, N2 - 7 106 Offset E mm 20.0 - Helix angle ϕ0 deg 35.0 Normal pressure angle φn0D, φn0C deg 22.7, 17.8 Radius of design point R20 mm - 101.0 Pitch cone angle γpw, Γgw deg 3.8 86.2 Top land of rack tcn mm 1.5 Clearance of rack cr mm 1.0 Gear addendum Ad2 mm - 0.5 Tooth depth on heel side mm 2.7 2.6 Tip diameter 2R1h, 2R2h mm 22.6 260.0 Face width b mm 35.0 30.0 Surface modification - none none

+∆P

u2

v2 z2

v1

z1

u1 +∆E

+∆G

+∆Σ +∆s

7

parameter



Fig. 7 Schematic of planar surface of action Fig. 8 Comparison of surface of action with that of face of SwD and SwC for drive and coast sides. gears. Conventional face and hypoid gears generally have curved surfaces of action, not planar surfaces of action.

Figure 8 shows the difference between the surface of action of the face and proposed hypoid gears. The face gear meshes on the curved surface of action. The normal direction is not constant along the surface of action of the face gear, whereas that of the proposed hypoid gear is constant as shown in Fig. 9. This feature would likely provide the advantage of low vibration because the direction of the mesh force is constant at all of the contact point, as with cylindrical involute gears.

Fig 9 The normal direction on the surface of action for drive side.

4.2 Conjugate Tooth Surface The plane of action SwD can be converted to the conjugate tooth surface of gear using Eq. (9), as shown in Fig. 10.

The intersection between SwD and the tooth surface is a line of contact with a certain rotational angle. The coast side is defined using the same method. The conjugate tooth surface of the pinion is also determined by Eq. (8).

Figure 11 shows a schematic of the proposed hypoid gear. The top land of the pinion decreases from the toe to the heel because of the use of parallel-depth teeth in the design. The tooth profiles of the pinion and gear in a cross section taken perpendicular to the pinion axis are shown in Fig. 12. The profile of the pinion is similar to that of an involute pinion of the face gear, and the gear has a nearly straight profile similar to that of a rack gear.

Planar surface for proposed hypoid gear

Curved surface for face gear

(a) Drive side

Toe

Heel (b) Coast side

Planar surface for proposed hypoid gear

Plane of action SwC

Plane of action SwD

Design point P0

Path of contact g0C

Path of contact g0D

Toe

Heel

Pitch line element

(a) Face gear (b) Hypoid gear

Toe

Heel

Tip

Root

u2c v2c

z2c

2

0

-2

-4

-20 -24

-16

100 120

140

Toe Heel

Tip

Root

u2c v2c

z2c

2

0

-2

-4

-20 -24

-16

100 120

140

8



Fig. 10 Transformation of conjugate tooth on drive side.

Fig. 11 Overall view of proposed hypoid gear.

Fig. 12 Cross section of pinion axis at u2 = 115.0 [mm].

Path of contact g0D Design point P0

Plane of action SwD

Conjugate tooth surface (gear)

Line of contact

z2

v2

Pinion

Gear

9



4.3 Line of contact The proposed hypoid gear meshes in the plane of action. Thus, the lines of contact are straight, as shown in Fig. 13.

The normal direction to the line of contact at each contact point, which is defined as the direction of the path of contact, is the same for all contact points.

In contrast, the lines of contact of a conventional face gear are curved. Thus, the direction of the path of contact varies at each contact point along the curved lines of contact because the surfaces of action of the face gear are curved.

Fig. 13 Lines of contact described with the tooth surface of gear at a certain rotational angle. The projection drawing of the gear tooth surface in a plane is shown in Fig. 14. The lines of contact are plotted at

intervals of one pitch. The proposed hypoid gear can simultaneously mesh three or four teeth, as shown by the four lines of contact in Fig. 14. Whereas, the lines of contact and paths of contact for the face gear are shown in Fig. 15. The difference of the number of the line of contact between drive and coast sides are larger for the face gear than for the proposed hypoid gear.

The contact ratios of the proposed hypoid gear and the face gear are given in Table 3. The contact ratios of the drive and coast sides are more similar to each other than are those of the face gear. This feature of the proposed hypoid gear could provide more similar meshing characteristics for drive and coast operations, as described in a previous study (Miyamura et.al, 2013).

Fig. 14 The projection of the lines of contact and paths of contact for proposed hypoid gear.

Drive side tooth

Line of contact

Coast side tooth

Toe

Heel

SwD

Toe

Heel Drive side tooth Coast side tooth

Line of contact

SwC

(a) Drive side (b) Coast side


Radial position �𝑢𝑢𝑟𝑟2𝑐𝑐2 + 𝑣𝑣𝑟𝑟2𝑐𝑐2 [mm]

Prof

ile p

ositi

on z r2

c [mm

]

Tip of gear

Tip of pinion

Toe Heel

Line of contact Path of contact g0D

Design point P0

1 pitch

100 130

0

2

−2

−4

110 120

Path of contact g0C

P0

Toe Heel

Tip of gear

Tip of pinion

1 pitch

Prof

ile p

ositi

on z r2

c [mm

]

0

2

−2

−4

Radial position �𝑢𝑢𝑟𝑟2𝑐𝑐2 + 𝑣𝑣𝑟𝑟2𝑐𝑐2 [mm] 130 100 120 110

10



Fig. 15 The projection of the lines of contact and paths of contact for face gear.

Table 3 Contact ratio. Proposed

hypoid gear Face gear

Drive Coast Drive Coast Profile 0.72 0.77 0.60 0.68 Lengthwise 3.18 3.21 3.56 3.07 Total 3.26 3.30 3.61 3.14

5. Numerical examination of influence on misalignment The tooth flank error δLm can be calculated using Eq. (12) to clarify the influence of meshing in the plane of

action. The error δLm is shown in Fig. 16 as contour lines with a light color for large value. The horizontal and vertical axes in Fig. 16 show the radial and profile positions of the gear tooth as is the case with Fig. 14. The translational misalignments ∆E, ∆P, and ∆G never affect the tooth bearing of the proposed hypoid gear; this is also an advantage of using cylindrical involute gears. The rotational misalignments ∆Σ and ∆s cause tooth flank error. These misalignments can induce edge contact because of the incline of the tooth flank error in the direction of the face width, as shown in Fig. 16.

Figure 17 shows the influence of the amount of each type of misalignment on the total tooth flank error for both the proposed hypoid gear and the conventional face gear. The total tooth flank error is calculated by subtracting minimum error from maximum error. The translational misalignment ∆P is not described in Fig. 17 because it does not affect the tooth bearing of the face gear or the proposed hypoid gear. Although the translational misalignments ∆E and ∆G of the pinion offset and gear axis produce tooth flank error in the face gear, they have no influence on the proposed hypoid gear as shown in Fig. 17(a).

The numerical results of translational misalignments are discussed in more detail. When ∆P, ∆E,or ∆G are given in Eq. (11) respectively, the tooth flank error δLm can be expressed by Eq. (12) as follows for Σ = 90 deg.

20 20

20 20

20

cos coscos sinsin

P b

E b

G b

Lm PLm ELm G

δ ϕ cδ ϕ cδ ϕ

∆

∆

∆

= ∆= ∆= ∆

(13)

Since the normal directions ϕb20 and c20 are constant for the proposed hypoid gear as shown in Fig. 9(b), the amount of δLm at each contact point is always constant in Eq. (13). This relationship supports the results of Figs. 16 and 17 (a).



Prof

ile p

ositi

on z r2

c [mm

]

130 100

0

2

−2

−4

120 110

Path of contact g0C

P0

Toe Heel

Tip of gear

Tip of pinion

Contact area

1 pitch Line of contact


Prof

ile p

ositi

on z r2

c [mm

]

100 130

0

2

−2

−4

110 120

Tip of gear

Tip of pinion

Toe Heel

Line of contact Path of contact g0D

Design point P0

Contact area

1 pitch

11



On the other hand, the amount of δLm for the face gear at each contact point differs because the normal direction of the face gear is not constant as shown in Fig. 9(a). This leads the tooth flank error by ∆E and ∆G on the face gear. As the tooth flank error of the face gear is not influenced by ∆P, the right side of Eq. (13) for δLm∆P also shows constant for the face gear.

Fig. 16 Tooth flank error with misalignment.

Fig. 17 Total tooth flank error of proposed hypoid and conventional face gears.

Toe

Tip of gear

Heel

−δLm [mm]

Tip of pinion (root of gear)

∆E = ∆P = ∆G = +0.1

∆Σ = +0.1

∆s = +0.1

Large error (contact zone)


No error

10 µm

0

1 µm

Toe Tip of gear

Heel

Tip of pinion (root of gear)

∆E = ∆P = ∆G = +0.1 −δLm [mm]

No error



∆Σ = +0.1

∆s = +0.1

0

10 µm

1 µm


(a) Translational misalignments (b) Rotational misalignments

0

0.05

0.1

0.15

0.2

-0.3 -0.2 -0.1 0 0.1 0.2 0.3Misalignment [mm]

Tota

l too

th fl

ank

erro

r [m

m]

Face(∆G) Face(∆E) Hypoid

(∆E, ∆G)

0

0.05

0.1

0.15

0.2

-0.3 -0.2 -0.1 0 0.1 0.2 0.3Misalignment [deg]

Face(∆Σ)

Face(∆s) Hypoid(∆Σ)

Hypoid (∆s) To

tal t

ooth

flan

k er

ror [

mm

]

12

Figure 16 shows that rotational misalignments ∆Σ and ∆s give inclined tooth flank errors. This result can be explained by Eq. (14) which shows the tooth flank error δLm lead by ∆Σ or ∆s. A point in which Eqs. (13) and (14) differ is that Eq. (14) consists of the tooth coordinates (u2, v2, z2). Even though the normal directions are constant for the proposed hypoid gear, the tooth flank errors δLm∆Σ and δLm∆s can be affected by the changes of contact point (u2, v2, z2). Thus, the tooth flank errors on the tooth are inclined in Fig. 16.



( )( )

2 20 2 20 20

2 20 2 20 20

sin cos cos

sin cos sinb b

b b

Lm u z

Lm v zs

δ ϕ ϕ c

δ s ϕ ϕ c∆Σ

∆

= ∆Σ −

= ∆ − (14)

The tooth flank error caused by rotational misalignment is lower for the proposed hypoid gear than for the

conventional face gear as shown in Fig. 17(b). The difference between the errors of the two gears is caused by their different surfaces of action. The plane of action for the proposed hypoid gear is advantageous because of the constant normal direction. For the design of some surface modifications, the characteristics of the proposed hypoid gear can reduce excessive crowning.

6. Conclusion A hypoid gear that meshes in the plane of action was developed using a novel design procedure. In this procedure,

the planes of action are defined first. Conjugate gear teeth are then obtained via coordinate transformations. The approach has been verified through some numerical examples. The obtained numerical results demonstrate the

hypoid gear meshing in the plane of action was available. The proposed hypoid gear has equal contact ratios for the drive and coast sides. Moreover, misalignments have less influence on the tooth flank error of the proposed hypoid gear than on that of a conventional face gear. These features need to be verified by experimental results in the future works.

References Aiki, K., Suzuki, A., Sugiura, H., Mizuno, T. and Hashimoto, M., Development of thin In-Wheel-Motor unit for EVs,

Transactions of Society of Automotive Engineers of Japan, Vol.48, No.2 (2017), pp.329–335 (in Japanese). Alves, J. T., Guingand, M. and Vaujany, J.-P., Designing and manufacturing spiral bevel gears using 5-Axis Computer

Honda, S., Design of hypoid gears having equal lengthwise contact ratios of drive and coast sides, Proceedings of MPT 2009-Sendai (2009), pp.70–75.

Honda, S., A New Tooth Geometry and its Applications, Soeisha, pp. 4–12 (2016a), pp. 31–93 (2016b), pp. 120–131 (2016c), pp. 94–115 (2016d).

Inoue, T. and Kurokawa, S., Derivation of path of contact and tooth flank modification by minimizing transmission error on face gear, Journal of Advanced Mechanical Design, Systems, and Manufacturing, Vol.6, No.1 (2012), pp.15–22.

Kawasaki, K., Tsuji, I. and Gunbara, H., Machining and running test of high-performance face gear set, Proceedings of the ASME 2011 IDETC/CIE, Paper No. DETC2011-48824 (2011), pp.73–80.

Litvin, F. L., Gonzalez-Perez, I., Fuentes, A., Vecchiato, D., Hansen, B. D. and Binney, D., Design, generation and stress analysis of face-gear drive with helical pinion, Computer Methods in Applied Mechanics and Engineering, Vol.194 (2005), pp.3870–3901.

Miyamura, H., Shibata, Y., Inagaki, M. and Aoyama, T., Design method for optimizing contact ratio of hypoid gears, Proceedings of the ASME 2013 IDETC/CIE, Paper No. DETC2013-12761 (2013).

Ohshima, F. and Yoshino, H., Study on high reduction face gears (1st report, modeling analysis), Transactions of the Japan Society of Mechanical Engineers, Series C, Vol.72, No.720 (2006), pp.336–342, (in Japanese).

Stadtfeld, H. J., Gleason Bevel Gear Technology, The Gleason Works (2014), pp.177–188. Tsuji, I., Gumbara, H., Abe, Y., Kawasaki, K. and Takami, A., Face gear with appropriately modified tooth flank,

Proceedings of MPT2009-Sendai (2009), pp.76–81.

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Design of high-reduction hypoid gears meshing in plane of ... · From this viewpoint, the design of the tooth surface can be independent from the tooth cutting process. Thus, the