design, generation and tooth width analysis of helical

Bulletin of the JSME

Journal of Advanced Mechanical Design, Systems, and ManufacturingVol.9, No.5, 2015

Paper No.15-00164© 2015 The Japan Society of Mechanical Engineers[DOI: 10.1299/jamdsm.2015jamdsm00

0123456789

66]

Design, generation and tooth width analysis of helical

curve-face gear

* The State Key Laboratory of Mechanical Transmission, Chongqing University 400044, Chongqing, China

E-mail: [email protected]

Abstract

A new type of gear pair consisting of a helical non-circular gear and a curve-face gear is designed, which can

transmit the time-varying motion of intersected axes of rotation. Based on the generation mechanism of

non-circular gear, the geometric model of helical curve-face gear is considered and the generating method of

pitch curve of helical curve-face gear is discussed. According to the meshing theory, the meshing equation of

helical curve-face gear is built and the mathematical model of tooth surface is developed. The method used to

calculate its minimum inner radius and maximum outer radius is obtained. Furthermore, the method that the

helical curve-face gear is generated by a shaper cutter is put forward and the three-dimensional model of

helical curve-face gear is completed by modeling software. A phenomenon of transmission stability is

discovered through contrast to the transmission ratio curve of experimental and theoretical value, which can

illustrate the correctness of the design method and the practicability of the helical curve-face gear.

Key words : Design, Helical non-circular gear, Face gear, Variable transmission ratio, Tooth width

1. Introduction

Face gear pair is a drive of meshing between involute cylindrical gear and face gear, commonly used to transfer

motion and force of intersected axes. Since the peculiarity of face gear pair in shunt and confluence drive, it shows

potential advantage in the application in the field of aeronautics and astronautics (Lewicki, et al., 2000, Heath and

Bossler, 1993, Filler, et al., 2002). Face gear drive system is usually applied in helicopter transmission system, relative

to the original drive system, it not only brought a massive relief on the weight, but also improved the reliability of the

transmission. Compared with the tradition bevel gear, face gear has many advantages, such as it can’t be affected by

axial installation misalignment, the contact ratio is larger and it has smaller vibration and lower noise (Gabiccini, et al.,

2004).

Litvin et al. investigated the meshing theory of gear transmission, obtained the condition of undercutting and

tip-cutting from the perspective of geometry principle, then solved the design of tooth width of face gear (Litvin and

Fuentes, 2004), and studied the face gear enveloping by the spur gear, helical gear and worm gear respectively. The

design of helical face gear, stress analysis and other fields were researched by Litvin (Litvin, et al., 2005). In recent

years, Chongqing University, Yanshan University and BUAA also did some research about face gear (Yang, et al.,

2010, Tang and Liu, 2012). Professor Zhu Rupeng et al. from NUAA developed a series of relevant theory of face gear

drive (Li and Zhu, 2008), they further ameliorated the design of tooth width of face gear from the perspective of the

change of meshing angle in the process of face gear generated by shape cutter (Zhu and Gao, 1999). Professor Fang

Zongde et al. from NWPU developed the molding, the condition of undercutting and tip-cutting of helical face gear

(Shen, et al., 2008). Professor Lin Chao et al. put forward curve-face gear pair according to the non-circular gear and

face gear (Gong, 2012, Lin, et al., 2013).

Since the specificity of face gear itself, it is easy to emerge the phenomenon of undercutting and tip-cutting on the

tooth of face gear. Therefore, under the condition of small transmission ratio, the tooth width of face gear is restricted,

and affects the carrying capacity of face gear. Thus, the change of the tooth profile or mending tooth shape will become

the new study direction of face gear. Curve-face gear also remains the same problem of tooth width limited. Based on

the design method of non-circular gear (Liu, et al., 2015, Shi, et al., 2012, Xia, et al., 2014), in order to further improve

Chao LIN* and Dong ZENG*

Received 10 March 2015

1

2

Lin and Zeng, Journal of Advanced Mechanical Design, Systems, and Manufacturing, Vol.9, No.5 (2015)

© 2015 The Japan Society of Mechanical Engineers[DOI: 10.1299/jamdsm.2015jamdsm0066]

the strength and stability, increase the contact ratio of curve-face gear and meet the requirements of high-speed

overloading in engineering, this paper designs the helical curve-face gear pair and develops its geometric and

mathematical model, so that it can have more potential applications in the field of agricultural engineering, textile

industries, aeronautics and astronautics et al.

2. The pitch curve of helical curve-face gear

2.1 The generation of pitch curve

The generation of pitch curve of helical curve-face gear can be considered as following process: a) the pitch curve

of rack is generated by conjugating with the pitch curve of helical non-circular gear. b) Enwinding the pitch curve of

the rack around a pitch cylinder with radius R can result in a spatial curve as shown in Fig. 1.

1- The pitch curve of helical non- circular gear 2- the pitch curve of helical curve-face gear

3- The pitch curve of non- circular rack

Fig.1 The generation of pitch curve of helical curve-face gear

As shown in Fig.1, Point 1P is on the pitch curve 1 and it will mesh with point

2P on the pitch curve 3 when

helical non-circular gear turns angle 1 . Enwinding the pitch curve of the rack around a pitch cylinder with radius R

can obtain the pitch curve 2, and the corresponding meshing point is '

2P . According to the spatial meshing theory, the

velocity of tangency point of the gear pair must be equal when pitch curves are meshing, namely 1 1 2r R .

1r is the equation of pitch curve of helical non-circular gear.

According to the meshing theory, the transmission ratio of helical curve-face gear can be derived as follows:

1

12

2 1

Ri

r

(1)

And in the process of transmission,

1 1

20 012

11 d r di R

(2)

2.2 The equation of pitch curve of helical curve-face gear

2O

2X

2Y

2ZR

1O

1X

1Y

1

2

3

1P

2P

M N

'M 'N

'

2P

1

2

2

2



Fig.2 The drive coordinate systems

As shown in Fig.2 the drive model of helical curve-face gear is established to show the meshing between helical

non-circular gear and helical curve-face gear. 1 1 1 1O X Y Z is the coordinate system which is rigidly connected with the

frame of the cutting machine, and coordinate 1' 1' 1' 1'O X Y Z is rigidly connected with helical non-circular gear.

Likewise coordinate 2 2 2 2O X Y Z is the coordinate system which is rigidly connected with the frame of the cutting

machine, and 2' 2' 2 ' 2 'O X Y Z is rigidly connected with helical curve-face gear.

1 and 2 refer to the direction of

rotation of the two gears respectively. When helical non-circular gear turns the angle 1 , and correspondently, the

helical curve-face gear turns the angle 2 . E is the distance between the axis of helical non-circular gear and the

bottom surface of helical curve-face gear. In the status shown in the Fig.2, (0)E r . R is the radius of the pitch

cylinder where the pitch curve of helical curve-face gear locates.

As shown in the Fig.2, the pitch curve of helical non-circular gear is a modified elliptical curve. According to the

theory of the modified elliptical curve (Litvin and Fuentes, 2004), the equation of arbitrary modified elliptical pitch

curve in polar coordinates can be represented as follows:

2

11 cos

a akr

k n

(3)

Where is the polar angle of a modified elliptical curve. The half of major axis and the deformation coefficient

of a modified elliptical curve are a and k , and 1n is the order of a modified elliptical curve.

On the basis of coordinate transformation theory, the transformation matrix used to transform the coordinate

1' 1' 1' 1'O X Y Z to the coordinate 2' 2' 2 ' 2 'O X Y Z can be represented as follows:

1 2 1 2 2 2

1 2 1 2 2 2

2'1' 2 '2 21 11'

1 1

sin sin cos sin cos cos

sin cos cos cos sin sin

cos sin 0 0

0 0 0 1

R

RM M M M

r

(4)

The coordinate values of meshing point is 1 1 1 1cos , cos ,0r r in the coordinate 1' 1' 1' 1'O X Y Z . Supposed

that the coordinate values of the point is 2' 2' 2 ', ,x y z in the coordinate 2' 2' 2 ' 2 'O X Y Z . According to the coordinate

transformation theory, there exist:

2' 2' 2' 2'1' 1 1 1 1 2 2 1, , cos , cos ,0 cos , sin , 0x y z M r r R R r r (5)

Based on the meshing theory, the point 2' 2' 2 ', ,x y z is on the pitch curve of helical curve-face gear, therefore, the

parametric equation of the pitch curve is:

2

2

1

cos

sin

0

x R

y R

z r r

(6)

Besides, the pitch curve of helical curve-face gear must be closed. So when 2 is in the range of [0,2 ] , the

3

1 1'O O

2 2'O O

1

2

2

1

R

1 1'Z Z

2 2'Z Z

1X

1'X

1Y

1'Y

2X

2'X

2Y

2'Y

E

2



order of helical curve-face gear 2n must be integer, namely it must satisfy the following equation:

1

2

02

2 1n r d

n R

，That is 1

2

2

02n

nR r d

(7)

Substituting Eq. (7) in Eq. (6) can result in the parametric equations of pitch curve of helical curve-face gear :

1

1

2

22

0

2

22

0

1

cos2

sin2

0

n

n

nx r d

ny r d

z r r

(8)

According to the equation of pitch curve above, the pitch curve can be drawn by Matlab as shown in Fig.3,

Fig.3 The meshing pitch curve with different parameters

According to the result as shown in Fig.3, following conclusions can be drawn: a) The module and the teeth

number of helical non-circular gear are positively correlated with the radius R of the pitch curve of helical curve-face

gear. b) Numbers of wave crest and wave trough of the pitch curve of helical curve-face gear can be controlled by 1n

and 2n , and the number of crest n can be expressed as

1 2n n n . c) As the deformation coefficient of helical

non-circular gear becomes larger, the crest value of the pitch curve of helical curve-face gear grows greater, so does the

non-uniform coefficient of 12i . With optimal design of parameters mentioned above, the pitch curve of helical

curve-face gear which is needed can be obtained.

-100

-500

50100

-100

-50

0

50

100-10

20

50

80

X/mmY/mm

Z/m

m

n1=2

n1=3

-100-50

050

100

-100

-50

0

50

100-10

10

30

50

70

X/mmY/mm

Z/m

m

k=0.1

k=0.2

1 1 24, 18, 0.1, 2&3, 2nm z k n n 1 1 24, 18, 0.1&0.2, 2, 2nm z k n n

4

3 The mathematical model of helical curve-face gear pair

3.1 Space meshing analysis of helical curve-face gear pair

1- Tooth profile of helical non-circular gear 2- Tooth profile of helical curve-face gear

3- Tooth profile of shape cutter - Pitch curve of helical non-circular gear

- Pitch curve of helical curve-face gear

Fig.4 The formation of the conjugate tooth profile

According to the formation of conjugate tooth profile and the space meshing theory, as shown in Fig.4, the tooth

Shape cutter

Helical non-circular gear

Helical curve-face gear P

I

II

1

2

3

2



profile 3 and the tooth profile 2 are the external meshing while the tooth profile 3 and the tooth profile 1 are the internal

meshing, so that the tooth profile 1 and tooth profile 2 can be enveloped by shape cutter at the same time. At any instant,

P is the common tangent point at which the pitch curve of shape cutter, the pitch curve and the pitch curve

keep respectively pure rolling contact. So the helical non-circular gear can guarantee meshing with helical curve-face

gear correctly at any time.

Fig.5 Analysis of the contact line

The helical curve-face gear obtained through the method in Fig.4 uses the method which the line contact is

replaced by point contact so that the contact traces are restricted to local. As shown in Fig.5, 1hL is the contact line

generated when shape cutter and helical non-circular gear are the internal meshing. 2hL is the contact line generated

when shape cutter and helical curve-face gear are the external meshing. M is the contact point when helical

non-circular gear mesh with helical curve-face gear.

In the process of transmission, the helical curve-face gear pair are the point contact and have longer contact line,

which can make it more stable in the meshing process. This design method above-mentioned can avoid to design

special cutting tools when the gear is machined. Furthermore, the method also can decrease installation sensitive

defect which is present in ordinary face gear.

3.2 Tooth and motion analysis of shape cutter

Fig.6 Cross-section of shape cutter Fig.7 The spatial position conversion of shape cutter

Based on the theory of involute helical gears, the tooth surface equation of shape cutter is represented as follows:

, cos sin

, , sin cos

h s s bs so s s s so s s

h s s h s s bs so s s s so s s

h s s s

x r

r y r

z p

(9)

Where bsr is the base radius.

s and so are the angle shown in Fig.6. corresponds to the left and right

tooth surface respectively. s is torsion angle of spiral motion and it is represented by tans k asu r .

asr is the

addendum circle radius and ku is a coefficient of tooth width.

sp is a spiral parameters.

M

2hL 1hL1hL

2hL

hO

'

hO

s

so

s

bsr

hX

hY

1O 1Y

1X

hOhY

hX

1hO

2hO

1hX

2hX

2hY

1hY

1L

1

1P

2P

tP

5

2



/ 2

/ 2

tan

s

so ns

ns ns ns

p H

Z inv

inv

(10)

Where the lead and the normal pressure angle are H and ns .

According to Eq. (9), the unit normal vector of tooth surface of shape cutter is represented as follows:

cos sin

cos cos

sin

b so s s

h b so s s

b

n

(11)

Where b is the base spiral angle and arctan( / )b bs sd p .

Using the generating method shown in Fig.4, the spatial positions conversion of shape cutter are shown in Fig.7.

The shape cutter shifts to the position 1 1 1h h hO X Y from the position

h h hO X Y , then the shape cutter rotates and

moves to the position 2 2 2h h hO X Y . a) The point

1P is the initial meshing point. b) Then it rotates angle

around the axis of shape cutter to point2P . c) The point

2P does pure rolling around the pitch curve of helical

non-circular gear to the pointtP . So

tP is the meshing point when the shape cutter locates at position2 2 2h h hO X Y .

By geometric relationships shown in Fig.7, the can be derived as

12 (12)

Where

arctan

'

r

r

.

According to the spatial position conversion process of shape cutter, the corresponding coordinate transformation

matrix can be obtained. So the coordinate transformation matrix used to transform the point 1P to the point

tP is

derived as

1 2 1 2 2 1 2 2

1 2 1 2 2 1 2 2

2'

1 1 1

sin sin cos sin cos sin sin cos

sin cos cos cos sin sin cos sin

cos sin 0 0 cos

0 0 0 1

h

L R

L RM

r L

(13)

Where 2 2

1 1 12 sinb bL r r r r , 2 2 2

1 1 1 1arccos 2bL r r L r . br is the pitch circle radius

of shape cutter.

3.3 Meshing equation and tooth surface equation of helical curve-face gear

In the meshing process of helical curve-face gear pair, the meshing equation can be represented as the product

between the unit normal vector 1

N and the relative velocity 1

V . The mathematical formula is as follows

1 1

1, , 0s sf N V (14)

Where the superscript denotes that parameters are in the coordinate 1 1 1 1O X Y Z .

By using coordinate transformation theory, the unit normal vector of left tooth surface is derived as

1

cos sin

cos cos

sin

b

b

b

N

(15)

Where 1 so s s . Correspondingly, the relative velocity of helical curve-face gear pair is

represented as

6

2



1 (1) (1) 1

1 2 3 1 1 2 2

1 1

1 1 21 21

21 1 1

,

sin cos sin

cos sin cos

sin cos sin

h h s s

bs bs s

bs bs s s s

bs bs s

V M r O O

r r L

r r L i p i R

i r r L

(16)

Where 21 121i i .

1 is the speed of helical non-circular gear.

Substituting Eq. (15) and Eq. (16) into Eq. (14) leads to the meshing equation of helical curve-face gear pair

1 1 21

21 1

, , cos cos cos cos cos

sin sin cos sin

0

s s bs b b b s s

b bs bs

f r L i p R

i r r L

(17)

According to the calculation method of tooth surface equation, the tooth surface equation of helical curve-face gear

can be derived as follows

2 2 1 2

2 2 1 2

1

1

sin sin cos cos sin sin

cos sin cos sin sin cos, ,

cos sin 0 cos

1

bs s s s

bs s s s

s s

bs s

r p R L

r p R Lr

r r L

(18)

4 Tooth analysis of helical curve-face gear

The phenomenon of tip-cutting and undercutting also will occur on helical curve-face gear when it is processed by

the shape cutter. The tip-cutting and undercutting will limit the tooth width of helical curve-face gear, which can affect

the carrying capacity and lead to failure of helical curve-face gear. Thus, the radius of tooth width of helical curve-face

gear needs to be optimized so that it can meet the design requirements.

4.1 Undercutting condition of helical curve-face gear

According to the undercutting condition in the gear geometry, the tooth surface of shape cutter h and the tooth

surface of helical curve-face gear 2 mesh with each other when helical curve-face gear is generated. In the same

coordinate system, the speed of shape cutter hv , the speed of helical curve-face gear 2v and their relative velocity

2hv will satisfy the condition : 2 2h hv v v .When 2 0v , the phenomenon of undercutting happens.

When the undercutting happens, there is an undercutting limit line which can be used to control the tooth surface of

shape cutter h and avoid the phenomenon of undercutting theoretically. According to the equation 2h hv v , they

satisfy the following equation:

1 11

2 0h s h sh

s s

r d r dv

dt dt

(19)

Where 1

hr represents the tooth surface equation of shape cutter in the coordinate 1 1 1 1O X Y Z .

Correspondingly the differential equations of meshing theory can be described as

1

1

0s s

s s

d d df f f

dt dt dt

(20)

Combine the Eq. (17), Eq. (19) and Eq. (20), the equation of the undercutting limit line can be represented by:

7

2



1

1

1 11

2

1

1

, , 0

0

0

h

s s

h s h sh

s s

s s

s s

r

f

r d r dv

dt dt

d d df f f

dt dt dt

(21)

According to Eq. (21), the linear equations can be derived as

1 11

2

1 11

2

1

1

0

h s h sh x

s s

h s h sh z

s s

s s

s s

x d x dv

dt dt

z d z dv

dt dt

d d df f f

dt dt dt

(22)

Where the component of (1)

hr in the X and Z direction are 1

hx and 1

hz . The component of (1)

2hv in the

X and Z direction are (1)

2h xv and (1)

2h zv .

Since the three equations in Eq. (22) are compatible equations, the rank of coefficient matrix M equals to the

rank of augmented matrix N , that is R M R N . Because of 2R M , the determinant of augmented

matrix N is equal to zero, that is 0N . Combined with Eq. (17), a nonlinear equation can be derived:

1 11

2

1 11

2

1

1

1

0

, , 0

h hh x

s s

h hh z

s s

s s

s s

x xv

z zv

df f f

dt

f

(23)

Correspondingly the undercutting limit points of helical curve-face gear are generated by the intersecting between

the addendum line of shape cutter and the undercutting limit line of helical curve-face gear. The tooth profile

parameters of shape cutter on tooth tip can be expressed as follows:

2 2

as bs

as

bs

r r

r

(24)

Where asr is the addendum circle of shape cutter.

According to Eq. (23) and Eq. (25), the undercutting limit points of helical curve-face gear in the coordinate

1 1 1 1O X Y Z can be expressed as 1 1 1

, ,h h hx y z

. So the minimum inner radius can be derived as follows:

2 2

1 1

1 h hR y R z

(25)

4.2 Tip-cutting condition of helical curve-face gear

With the radius of helical curve-face gear increasing, the left and right tooth surface will gradually close, and then

the phenomenon of tip-cutting will emerge. Tip-cutting will decrease the strength of helical curve-face gear. In order to

avoid the phenomenon of tip-cutting, controlling the maximum radius of helical curve-face is necessary.

8

2



Fig.8 Tip-cutting analysis

According to the condition of tip-cutting, the tooth thickness at the sharpened place is zero. As showed in Fig.8, the

helical curve-face gear is generated by shape cutter. h hO Z and

2 2O Z are the rotation axis of shape cutter and helical

curve-face gear, 1 1hO I is the instantaneous rotary axis at the moment. At the instant shown in Fig. 9, point P is the

intersection between pitch curve and instantaneous rotary axis. P is also the pitch point. Q is a moving point on the

instantaneous rotary axis. Assume that the section A A passes through point P , and the section B B passes

through point Q as shown in Fig. 9:

(a) (b)

Fig.9 Section analysis

As shown in Fig. 9(a), there exists no tip-cutting phenomenon at the place of section A A but there exists at the

place of section B B .Suppose that the left and right tooth profile of helical curve-face gear are straight lines and the

section B B is the place where the phenomenon of tip-cutting appears. So the maximum outer radius is the sum of

the radius of pitch curve of helical curve-face gear R and the distance between the two sections l .

In the Fig. 9(b), suppose that M is the meshing point. Thus the point M is on the surface of tooth. The

coordinate value of M can be expressed as:

cos sin , sin cosbs so s s s so s s bs so s s s so s sM r r

The tip-cutting appears at the tooth tip of helical curve-face gear, which is generated by the tooth root of shape

cutter in the generating process. So K is the point on the dedendum circle of shape cutter and K is also on

coordinate axis hX of shape cutter. Correspondingly the coordinate value of K can be expressed as:

,0bs fK r h

According to the geometric relationship shown in Fig. 9, it leads to the following equations:

sin costan

cos sin

bs so s s s so s s

bs so s s s so s s bs f

r

r r h

(26)

Where so s ，cos

2cos

h n t

bs

z mr

，

tantan

cos

n

t

， f n an nh m h c ， 20n ， 1anh ， 0.25nc 。

Suppose the solution of Eq. (26) is and according to Fig. 8 and Fig. 9, the geometric equation can be derived

Shape cutterHelical curve-face

gear

R

hZ2Z

2O

hO1hO

1I

A

A

B

Bl

P

Q

hO

A A

bsr

P

nbr

hX

hY

hOB B

hX

hY

bsr

MK

Q

tooth profile of

helical curve-face

gear

tooth profile of

shape cutter

9

2



as:

cos cosk k

bs bs

O Q O P

t

r rh L L

(27)

sinb

hRl

r

(28)

According to Eq. (27) and Eq. (28), the maximum outer radius 2R of helical curve-face gear satisfies the

following equation:

2sinb

hRR R l R

r

(29)

4.3 The tooth width of helical curve-face gear under different parameters

The controlling variable method is adopted as an approach to study the influence on the tooth width of helical

curve-face gear with different parameters. The original parameters is as follow: Helical curve-face gear order is

2 2n ; Helical non-circular gear order is 1 2n , the deformation coefficient is 0.1k , the tooth number is

1 18z ;

Normal module of shape cutter is 4nm , the tooth number is 12hz , the normal pressure angle is 20n and

the helix angle is 20 . Calculate the radius 1R and

2R of helical curve-face gear with different parameters by

Matlab and draw the curve of tooth width as shown in Fig.10.

Fig. 10 The influence of gear parameters on the undercutting and tip-cutting radius

As shown in Fig.10, there is a positive correlation between 1n , 2n , and the maximum outer radius 2R and

the minimum inner radius 1R of the helical curve-face gear. There is a negative correlation between the deformation

coefficient k and the maximum outer radius 2R and the minimum inner radius 1R of the helical curve-face gear.

So by controlling and optimizing the gear parameters, the condition of undercutting and tip-cutting can be satisfied and

the helical curve-face gear satisfied the design requirements will be obtained.

Since the teeth of helical curve-face gear distributed on the curved surface, the tooth profile of every tooth is

different and each tooth has different maximum outer radius and minimum inner radius. In order to guarantee the

helical curve-face gear avoids the phenomenon of tip-cutting and undercutting, the tooth width limited condition of

each tooth should be satisfied. Thus, the maximum outer radius becomes smaller and the minimum inner radius

1 2 3 430

60

90

120

150

180

The orders of helical non-circular gear n1

Th

e ra

diu

s R

1 a

nd

R2

The maximal outer radius R2

The minimal inner radius R1

1 2 3 430

60

90

120

150

180

The orders of helical curve-face gear n2

Th

e ra

diu

s R

1 a

nd

R2

The maximal outer radius R2

The minimal inner radius R1

0.05 0.1 0.15 0.2 0.25 0.380

85

90

95

Th

e m

axim

al o

ute

r ra

diu

s R

2

The eccentricity of helical non-circular gear k

0.05 0.1 0.15 0.2 0.25 0.360

65

70

75

Th

e m

inim

al i

nn

er r

adiu

s R 1

R2

R1

8 10 12 14 16 18 2084

86

88

90

Th

e m

axim

al o

ute

r ra

diu

s R 2

The helix angle of shape cutter /。

8 10 12 14 16 18 2066

68

70

72T

he

min

imal

in

ner

rad

ius

R 1

R2

R1

10

2



becomes bigger. According to calculations (Shen, et al., 2008), under the same parameter condition, the tooth width of

helical curve-face gear is smaller than the tooth width of the helical face gear.

5 Three-dimensional modeling and comparative analysis of helical curve-face gear

5.1 Three-dimensional modeling of helical curve-face gear

Based on the study of the forming and optimal design principle of helical curve-face gear, the three-dimensional

model of helical curve-face gear can be obtained by using the parameters shown in the Table.1.

Table.1 Parameters of helical curve-face gear pair

helical non-circular

gear

helical curve-face

gear

Tooth number of shape cutter 12hz

Normal pressure angle of shape cutter 20n

Normal module 4nm mm

Hand of helix Left-hand Right-hand

Helix angle 10

Order 1 2n

2 2n

Deformation coefficient 0.1k /

Tooth number 1 18z

2 36z

Major axis 35.817a mm /

Cylindrical surface radius of pitch curve / 72.386R mm

Inner radius 1 70R mm

Outer radius 2 83R mm

As shown in Fig.11, there are 4 wave crest and 4 wave trough on the 3D model of helical curve-face gear. The teeth

distribute uniformly on the surface. This structure not only can meet properly mesh with helical non-circular gear but

also can improve the contact strength and bending strength and increase the contact ratio and meshing stationarity on

the basis of face gear.

1- Helical non-circular gear 2- Helical curve-face gear

Fig.11 3D model of helical curve-face gear pair

5.2 Transmission ratio analysis

Based on the 3D model obtained, the transmission ratio of helical curve-face gear can be comparative analysis with

the transmission ratio of straight curve-face gear. A comparison of theoretical transmission ratio of helical curve-face

gear with theoretical and experimental transmission ratio of straight curve-face gear is made in Fig. 12. The parameters

of straight curve-face and the experimental parameters are shown in the Table. 2.

Table.2 The parameters of the experiment

Parameters Value

Order of straight non-circular gear 2

12

11

2



Order of straight curve-face gear 2

Tooth number of straight non-circular gear 18

Tooth number of straight curve-face gear 36

Modulus (mm) 4

Tooth width (mm) 13

Input speed (r/min) 100

Load torque (Nm) 20

The experiment device can be established as shown in Fig.12. Based on the parameters shown in the Table. 2, the

experimental results can be shown in Fig.13.

Fig.12 The experiment device of straight curve-face gear

Fig.13 Comparative analysis of transmission ratio

As shown in Fig.13, The cycle of transmission ratio of helical curve-face gear is the same with straight curve-face

gear’s, and the value changes between 1.6 to 2.2. There is no catastrophic phenomenon on the wave crest and wave

trough of the transmission ratio curve of helical curve-face gear, which can indicate the helical curve-face gear pair

drive more smoothly. Compared with the experimental values, the cycle of transmission ratio of helical curve-face gear

is much clearer and the change in value is more stable. Thus, the helical curve-face gear pair are more conducive to

drive rotation motion of intersected axes and to meet the needs of engineering practice.

6. Conclusion

1) A new type of curve-face gear drive rotation motion of intersected axes has been investigated. The gear drive is

Driving motor

Load motor

Straight curve-face

gear pairGearbox Torque speed

sensor

0 0.5 1 1.5 2 2.5 31.6

1.8

2

2.2

Time/s

Tra

nsm

issi

on

rat

io

Theoretical transmission ration of straight curve-face gear

Experimental transmission ration of straight curve-face gear

Theoretical transmission ration of helical curve-face gear

5 Cycles

12

2


© 2015 The Japan Society of Mechanical Engineers[DOI: 10.1299/jamdsm.2015jamdsm00 ]66

formed by a helical non-circular gear and a helical curve-face gear.

2) Based on the presented formation mechanism of pitch curve, the generalized design method of pitch curve of

helical curve-face gear has been obtained. The meshing equation and tooth surface equation of helical curve-face gear

have been developed.

3) Computerized design of the helical curve-face gear drive that avoids tip-cutting and undercutting has been

developed. 3D model of helical curve-face gear pair has been generated.

4) Through contrast to the transmission ratio curve of experimental and theoretical value, the correctness of the

design and the transmission stability of helical curve-face gear have been verified.

Acknowledgment

The authors would like to appreciate their supports from the National Natural Science Foundation of China

(51275537).

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design, generation and tooth width analysis of helical

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