research on the geometric modeling and tooth-flank

Bulletin of the JSME

Journal of Advanced Mechanical Design, Systems, and ManufacturingVol.10, No.2, 2016

Paper No.16-00130© 2016 The Japan Society of Mechanical Engineers[DOI: 10.1299/jamdsm.2016jamdsm0033]

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Abstract Based on the space engagement theory and cam theory, an approach of the tooth generation of curve-face gear, concerned the tooth profile equation, tooth characteristics analysis, manufacture and error evaluation is obtained. The tooth surface equation of curve-face gear is obtained and the tooth surface is produced by inserting advanced surface command in SolidWorks combining with the interference analysis. An error evaluation method, based on the extracting points from a series of sections along the tooth length, is put forward. The tooth-flank characteristics, concerned the distribution angle and meshing angle, are analyzed and the interference phenomenon is discussed. Finally, the validity of the theory is demonstrated by the rolling experiment and additive manufacturing.

Key words : Curve-face gear, Time varying, CAM mechanism, Tooth flank characteristic, Geometric modeling

1. Introduction

The face gear drive is composed of a cylindrical gear and a bevel gear with mutual engagement (Hou and Zhu,

2009).However, the rotational speed of the output is constant, which is unable to satisfy some special occasions (Litvin et al., 1994). Due to these shortages, a new space translation type of conjugated gear, whose output speed changes according to specific function, is presented. Curve-face gear, with a high carrying capacity, light weight and combining the characteristics of the cylindrical gear and bevel gears, can be used to transfer motion and power between the intersecting and staggered shaft.

The concept of face gear, which is defined as a series of racks with variable pressure angle, is firstly proposed by Buckingham in 1940; Large amounts of researches about strength, TCA, contact ratio and main curvature of surface are done by Nanjing university of aeronautics and astronautics (Jin et al., 2010); A new type of face gear called curve-face gear is proposed by Chongqing university, and the surface modeling, simulating manufacturing, kinetic characteristics and measurement method are analysis(Lin et al., 2013, 2015a, 2015b).

Curve-face gear drive is a non-standard gear type. Given the complexity of the design method, especially the tooth surface modeling and manufacturing methods. These defects greatly restrict the development of this gear pair. Generally, the research of the non-circular gear mainly focuses on the solution of tooth profile (Li and Wu, 2005. Tong et al., 2013). The tooth profile of non-circular bevel gear was obtained from the view of spherical coordinates by BeiHang university (Xia et al., 2008). Based on the cylindrical coordinates and the space engagement theory, the arbitrary curve equation of curve-face gear was obtained from the perspective of cylindrical coordinate system and the expansion method was established (Lin et al., 2015c). The concept of meshing angle function was defined, and a direct profile design method for generating non-cylindrical gear pair was proposed by Northeastern university (Lin et al., 2003).

All the mechanical products, machine or instrument, which contain gear transmission, structure performance, working accuracy, bearing capacity and service life, are all related to gear precision (Litvin et al., 2004). Nowadays, there are mainly two measurement methods (Sato et al., 2010. Suh et al., 2002. Sanchez et al., 2012). One is the

1

Chao LIN *, Zhiqin CAI* and Yao WANG** * The State Key Laboratory of Mechanical Transmission, Chongqing University

131 Yubei Rd, Shapingba, Chongqing, China

E-mail: [email protected]

** Taiyuan University of Science and Technology

030024, Taiyuan, China

Received 26 February 2016

Research on the geometric modeling and tooth-flank characteristic of curve-face gear

2

Lin, Cai and Wang, Journal of Advanced Mechanical Design, Systems, and Manufacturing, Vol.10, No.2 (2016)

© 2016 The Japan Society of Mechanical Engineers[DOI: 10.1299/jamdsm.2016jamdsm0033]

geometric error measurement with the help of the gear measuring equipment; Another one is the contact pat-tern detection with a rolling tester checking the gear pair contacts spot. Similarly, for the detection of non-circular gear, flank test are the mainly way and some scholars also propose to use coordinate measurements (Parey et al., 2012. Ahamed et al., 2014).

Based on the space engagement theory and cam theory, an approach of the tooth generation of curve-face gear, concerned the tooth profile equation, tooth characteristics analysis, manufacture and error evaluation, is obtained. The validity of the theory is demonstrated by the rolling experiment and additive manufacturing.

2. Principle

As shown in Fig.1, cylindrical cam is the driving element, whereas the roller and the sliding block are both the driven element. Roller and the cam are in tangency at point M. When the structure size of cam mechanism is confirmed, the rotating speed of roller and the movement speed of link are both the function of the rotating speed of cam. This kind of driven structure is defined as composite mechanism which can achieve the rotation of intersecting axles as well as the movement of output axle.

Fig.1 Cam mechanism

Define the cam mechanism as the conjugate movement between curve-face gear and non-cylindrical gear. The pitch curve of non-cylindrical gear is a composite curve, which is composed of the rotation of roller as well as the movement of sliding block as shown in Fig.2.

Fig.2 Pitch curve of non-cylindrical gear

As shown in Fig.2, is the rotation angle of roller; 1 is the rotation angle of non-circular gear; s is axial

displacement. They can be defined as

1

1

2 2

01

(0) ( )

( )( ) ( ) / g

s r r

drr r

d

(1)

where 1r is the pitch curve of non-cylindrical. gr is the radius of roller.

The pitch curve of curve-face gear is the trajectory of cylindrical cam, which can be defined as

2

1

2

cos( )

(0) ( )

sin( )

x R

y r r

z R

(2)

where 2 is the rotation angle of curve-face gear, R is the cylindrical radius of curve-face gear.

3. Tooth surface equation of curve-face gear

Generally, the manufacturing method of gear is based on the conjugate meshing between standard involute

2

2



cylindrical gear cutter and the gear blank, so does curve-face gear. However, the moving path of curve-face should obey the cam theory as well as the meshing theory.

Fig.3. Meshing relationship of curve-face gear pair

According to the movement relationship between non-cylindrical gear and curve-face gear, the meshing coordinate systems are established as shown in Fig.3. Coordinate OF-XFYFZF is rigidly connected to the frame of curve-face gear. Coordinate O2-X2Y2Z2 is rigidly connected to the curve-face gear, and the two coordinate systems are overlapped at the initial time; Coordinate ON-XNYNZN is rigidly connected to the frame of non-cylindrical gear. Coordinate O1-X1Y1Z1 is rigidly connected to the non-cylindrical gear; Coordinate OK-XKYKZK is rigidly connected to the frame of cylindrical gear cutter. The rotation angle of non-cylindrical gear and curve-face gear are 1 and 2 , respectively.

2

0

1

'

(0) ( )

E OO ei

S O O s j

s r r

(3)

where e is the distance between O2 and plane XOZ, s is the axial displacement of driving gear. The tooth profile of cylindrical gear cutter can be depicted as

cos

sin

K k k

k K k k

K k

x r

y r

z u

R

(4)

where kr is polar radius, k is polar angle, ku is tooth width parameter. Normal vector of driving gear can be depicted as

sin( )

cos( )

0

k kM k k

k kk M k k

k kM

k k

u aa u

v a

wa u

R R

nR R (5)

Convert kR and FR into fixed coordinate system O-XYZ and OF-XFYFZF.

2

1

2

2 1

j

k

J

Fe

R R S R

R R

R R E (6)

where FR is the tooth profile of curve-face gear.

Then the value of FR can be depicted as 2

2(( ) ( (0) ) ( ( )) )jJ

F K K Ke e x y r R z R R i j k (7)

Ultimately

2 2 2

2 2 2

2

( )cos ( ( ))sin

( )sin ( ( ))cos

(0) ( )

F K K

F F K K

F K

x e x z R

y e x z R

z y r R

R (8)

Derivative Eq.(8)

2 222 1 2

2

j jJ J

j F F k

dJ e e

d

R

R R R R S R (9)

then we can define

3

2



2

2

1

12 1 2

jJ

r F

r k

r r

e

V R

V R

V V V

(10)

Then

12 2 2(( ) ( ) ( ( )) )j K K Ks J x e s y z R V j i j k (11)

By the basic properties of the rotation transformation tensor

12 2 2( ( )) ( )K Kz R s x e V i j k (12)

Form Eq.(12), compared with face gear, the relative velocity of curve-face gear pair is variable, whereas face gear pair is a constant.

Then the meshing equation can be expressed as

k k 1 12( , , ) 0Kf u a V n (13)

Namely

2 2( ( )) ( ) 0k K k k Ku z R v s w x e

The equation can be deduced as

2tan( ) ( )k k ku a s R (14)

So the tooth surface equation of curve-face gear is

2 2

2 2

2

cos cos tan( ) sin

cos sin tan( ) cos

sin (0) ( )

k k k k

F k k k k

k k

e r a s

e r a s

r r R

R (15)

Compared with the previous method (Lin et al., 2013), large differential calculations are reduced in Eq.(15), which improves the calculation accuracy and efficiency.

According to the Eq.(15), it indicates that the basic parameters of tooth surface equation are 1 , ka and ku . ka

changes within the range of [aa, af], and the meshing angle 1 changes within the range of [ 1i , 2i ]. When dealing

with the numerical solution of curve-face gear, we can choose appropriate meshing angle and pressure angle as the known quantity at first, then reverse the values of uk. 4. Error evaluation of equation

The 3D model can be produced by inserting the discrete points (from Eq.(16)) to SolidWorks combining with the advanced surface command.

To explore the precision of tooth model, the tooth profile of the curve-face gear can be regarded as a series of racks with variable pressure angle along the tooth length (As shown in Fig.4). Assuming that the profile points extracting from the racks can satisfy the gear tooth surface equation.

Fig.4 The extracted method of profile points

Plug the extracted values (X*,Y*,Z*) into the tooth surface equation. The solving process is as shown in Fig.5. Then the basic parameters 1 , ka and ku of tooth profile equation can be obtained.

4

2



Fig.5 Calculation diagram of basic parameters

The error of normal vector values between theoretical tooth surface and the model can be defined as * * * *

1( , , ) ( , , )i i iF F F F F k k i Fx y z a u n R R n (16)

where in , (1, , )i k is the normal error of each extracted point. Fn is the unit normal vector of curve-face gear which

can be obtained as

1

1

=

F F

kF

F F

k

a

a

R R

nR R

(17)

5. Tooth surface characteristics 5.1 Tooth Profile Distribution

As shown in Fig.6(a), in order to ensure the meshing of curve-face gear pair, the teeth of curve-face gear should be uniformly distributed along the pitch curve.

(a) Distribution angle of curve-face gear (b) Arc length along the tooth length

Fig.6 Distribution rule of curve-face gear

However, due to the characteristic of variable tooth thickness, the corresponding arc length of each section is different as shown in Fig.6(b).

The arc length of each section along the pitch curve can be obtained as : :n nL L R R (18)

where L is the overall arc length of pitch curve which can be expressed as 2 2 2

t 10( ) ( )dL r r mz

(19)

where z1 is the teeth number of the non-cylindrical gear; m is the module. tr is the differentiation of polar radius.

Divide the arc length nL into 2n1 parts in one cycle (from the through to the peak of pitch curve within the range of [0,

1/ 2n ]), then the arc length of each tooth profile can be defined as

1/ 2'=n nLL z ,=1,2… 1 1/z n (20)

By Eq.(19)~(20), the distribution angle i of each tooth profile can be inverse calculated.

Compared with face gear, the distribution angle i of each profile along the pitch curve is not uniformly distributed.

The meshing angle 1i of each tooth profile can be expressed as

21

2

( )i i

i ii

yatg

x

1

21

( )iii i

ii

i

xatg a

y

(21)

5

2



where, iia is the pressure angle which can be determined by the following equation.

+2i i iu ia (22)

where, u is the pressure angle of involute cylindrical gear, and ii can be expressed as

1

1 1

cos 1=atan

sini

ii

i

e n

en n

(23)

B1i (x1i, y1i) is the actual enter meshing point and B2i (x2i, y2i) is the actual outer meshing point as shown in Fig.7.

Fig.7 Limit meshing point of curve-face gear pair

From Eq.(22) and (23), the meshing angle 1i and pressure angle iia of each tooth profile are different.

5.2 Tooth surface interference Due to the interference of tooth surface, which is similar to face gear (Jin et al., 2010), the tooth width of the

curve-face gear will be restricted. To avoid interference, the boundary points of root should be excluded. However, due to the periodical change of

the tooth profile of curve-face gear, though the intervention happens in one teeth, other tooth may not necessarily interference in the same position. So the general method has significant limitations. It is known that the interference is most likely located in the position of the minimum curvature radius. In this paper, the interference phenomenon will be discussed from the point of the main curvature.

The basic equation can be calculated as 2 2

1 1

0

k ka a

E F G

L M N

(24)

where E, F, G and L, M, N is the first basic amount and the second basic amount of cylindrical gear and curve-face gear, respectively, which can be obtained as

1 1

2 2 2

2 21 1

2 2F F F F

k k

F F F F F F

kk

( ) ( )a aE F G

=L M N

aa

R R R R

n R n R n R (25)

where

2 22

2 2

2 2 2 2

2 2 2 21

2 2

cos sin

tan sin cos

tan

sin cos sin

cos s

2cos,

in cos

sin 2 sin

0

( )

cos , tan(

os

)

c

F

kk

F

k

k k

k k k k

b k

k

k

k b k

A

r Aa

A

C Ds Ds

C Ds Ds

B

sA

r a

C e r D

a B

a

aB

a a

a

R

R

(26)

So the main curvature can be expressed as

6

2



2 21 1

2 21 1

2

2k k

k k

L M a Nd ak k

E F a G a

(27)

For curve-face gear, interference is focused on the addendum part of the tooth. Therefore, this paper take the tooth profile of the addendum part as the research object to explore how the curvature of tooth addendum influence the interference. 6. Simulation

Based on the theory above, the analysis of tooth surface modeling, error evaluation and tooth surface characteristic of the curve-face gear are done.

The basic parameters and machining parameters of curve-face gear are as follows in Tab.1. Table.1. Basic parameters

Curve-face gear Non-cylindrical gear

Modulus 4mm 4mm

Number of teeth 36 18

Order 2 2

Inside diameter 70 10.5

Outside diameter 83 44

Center distance 47.4mm

Minimum radius 1.1mm 2.89mm

Material 45# 40Cr

Blank size Φ166x30mm Φ88x22mm

The discretization points of curve-face gear are shown in Fig.8(a). The tooth surface was produced by inserting advanced surface command in SolidWorks combining with the interference analysis. Finally, the 3D model of curve-face gear is shown in Fig.8(b).

(a)Discretization of curve-face gear (b) 3D model of curve-face gear

Fig.8. Tooth generation of curve-face gear

As shown in Fig.9(b), the tooth of curve-face gear is not in a plane but uniformly distribute along the pitch curve. Combined with the theory of Fig.4, extract the profile points from the 3D model, then the basic parameter can be obtained as shown in Table.2.

Table.2. Basic parameters of tooth-1

Left tooth surface Right tooth surface

ak 1 uk ak 1 uk

0.636 3.087 -0.27 0.625 -0.18 2.555

0.627 3.081 -0.261 0.617 -0.18 2.56

0.617 3.075 -0.247 0.607 -0.178 2.566

0.605 3.068 -0.229 0.597 -0.17 2.572

0.594 3.062 -0.216 0.586 -0.16 2.578

… … … … … …

0.232 2.964 -0.059 0.223 -0.058 2.677

0.191 2.961 -0.058 0.187 -0.058 2.68

7

2



0.141 2.958 -0.055 0.147 -0.058 2.683

0.07 2.956 -0.04 0.062 -0.058 2.685

According to Eq.(16), the theoretical error can be obtained as shown in Fig.10. As shown in Fig.9, the error change rule of curve-face gear picked at the three different positions (inner, medial

and outboard flank) are similar. The maximum error of inner and outboard flank is 8um, which is very small, it is confirmed that the inner and outboard tooth profile are available. However, since the tooth surface of medial flank is generally generated by the scanning traces (the command of SolidWorks), the error of medial flank is relatively larger (14um).

Fig.9. The theoretical error of gear tooth profile

Main curvature of tooth surface can be obtained through the Eq. (27). And the meshing angle of each tooth profile can be got from Eq.(21). Due to k and k are equal and opposite, so only one value is analyzed.

(a) Main curvature of tooth surface (b) Main curvature of single tooth

Fig.10 Main curvature of curve-face gear

Fig.10 illustrate the variations of the main curvature of the tooth profile. In fact, the change rule of different sections is similar, so this paper just listed the curvature of the medial section.

As shown in Fig.10(a), 1) In one engagement period from trough to peak of the pitch curve, the main curvature increase slowly and reach the maximum value at tooth-1, which located at the through position of pitch curve; 2) In one tooth profile engagement period, the tooth profile curvature reduces from the dedendum to addendum of tooth profile. That is, the addendum of tooth-1 is most likely to interference.

Fig.10(b) illustrate the variations of the main curvature of single tooth along the tooth length. As shown in Fig.12, the main curvature reduces from inner flank to outboard flank, which can be verified in the Fig.9 (b). That is, the tooth profile of inner flank is mostly like to interference. 7. Experiment 7.1 Manufacturing and Measurement error

The entity of curve-face gear is processed by DMU 60 mono BLOCK five axis machining center as shown in Fig.11.

1 4 7 10 13 16 19 22 25 28 31 34 370

0.002

0.004

0.006

0.008

0.01

0.012

0.014

Extract point number

Com

posi

tion

err

or/(

mm

)

Tooth-1 Tooth-2 Tooth-3 Tooth-4

Tooth-5

Inner flank Medial flank Outboard flank

0 10 20 30 40 50 60 70 80 901.2

4.2

7.2

10.2

13.2

16.2x 10

-3

Rotation angle 1(°)

Cur

vatu

re R

-1(m

m-1

)

Right tooth profile Left tooth profile

Tooth-5

Tooth-2 Maximum

Tooth-1 Tooth-3 Tooth-4

55 60 65 70 75 80 85 900

0.003

0.006

0.009

0.012

0.015

Rotation angle 1(°)

Cur

vatu

re R

-1(m

m-1

)

Inner flankMedial flankOutboard flankLeft tooth

profile

Meshingangle

Meshing angle

Addendum

Dedendum

Right toothprofile

8

2



Fig.11 Manufacturing process

The manufacturing error are obtained by the contour scanning software of the German Colin berg P26 CNC Gear Measuring Center as shown in Fig.12.

Fig.12 Measurement of the curve-face gear artefact

Set the pitch curve of curve-face gear as a benchmark firstly. Then, a series of points which are enveloped by the involute part of cutting tool can be obtained as shown in Fig.13.

Fig.13 The points of involute part of curve-face gear

Refer to the calculation method of theoretical error (Eq.(16)), the manufacturing error of three positions can be obtained as shown in Fig.14.

The relative average error can be defined as

1

1 N

Nn nN

(28)

From Fig.9, the theoretical average error is 7um, whereas the manufacturing error is 9um (Fig.14). That is, although the measurement error composed of theoretical error and manufacturing error caused by machining, heat treatment and other reasons, the theoretical error account for most of the proportion. So the manufacturing method is available.

Fig.14 The measurement error of curve-face gear

7.2 The rolling experiment and additive manufacturing

-50-45

-40-35

-30-25

-20-15

-10-5

05

-75-70

-65-60

-55-50

-6-30369

x(mm)y(mm)

z(m

m)

Right tooth profileLeft tooth profile

Dedendum curve

Tooth-1

Tooth-2

Tooth-3

Tooth-5Addendum curve Tooth-4

1 5 9 13 17 21 25 29 33 360

0.003

0.006

0.009

0.012

0.015

0.018

0.0210.021

Measure point number

Com

posi

tion

err

or/(

mm

)

Inner flank Medial flank Outboard flank

Tooth-1 Tooth-2 Tooth-3Tooth-4

Tooth-5

9

2



(1) The rolling experiment In order to verify the correctness of the tooth profile equation. The rolling experiment is necessary. The rolling

experiment are conducted on a 500 mm universal rolling inspection machine as shown in Fig.15 (a). A pair of curve-face gear with display agent is installed on two main shaft of the machine, respectively. In order to guarantee the correct meshing of the gear pair, the mounting distance must be adjusted and the gear pair can roll normally under the driving of the motor.

1#

2#3#

4#5#

(a) Rolling experiment (b) Contact status

Fig.15. The rolling experiment of curve-face gear

As shown in Fig.15 (b), the rolling experiment of the processed curve-face gear pair indicates that the contact area is distributed on the whole tooth surface. In order to evaluate the contact quality of curve-face gear pair, the contact area percentage along the breadth of the tooth and the depth of tooth can be obtained as shown in Fig.16.

Contact line

'L

L

''L

H'H

Fig.16. The definition of contact area percentage

As shown in Table.3, though the contact quality along the breadth of the tooth is higher than that of the depth of tooth, both of the contact area percentage are more than 50%, which verified the correctness of the theory.

Table. 3. Evaluation of contact quality

Tooth

Labelled

Percentage along breadth of tooth ' '' / 100%L L L L

Percentage along depth of tooth ''/ 100%H H H

1# 96.2 51.1

2# 91.6 64.3

3# 86.0 68.2

4# 84.2 75.2

5# 87.3 72.4

(2) The interference analysis Tooth width, which effects the intensity of tooth root, is one of the main factors in calculation of curve-face gear,

and tooth surface interference is the main factor which limits the tooth width. In order to verified the correctness of tooth profile interference theory. The interference entity can be obtained by the additive manufacturing as shown in Fig.17. The reason for choosing this manufacturing method mainly focused on 1) the existing generating processing methods are not suit for curve-face gear, and recently only the method of the numerical control machining and additive manufacturing can be applied. 2) Due to the interference of the tooth surface, which can easily result in the tool wear and damage, the additive manufacturing method is a better choice.

10

2



Fig.17 Additive manufacturing

The additive manufacturing machine chosen for the processing of the gear is EOS M280, the basic processing principles are 1) the processed model can be obtained by slicing CAD models in Color STL or ESTL format. 2) Before the processing, the molding chamber has to be preheated and the manufacturing will start when the oxygen content in it is lower than a certain value.3) A single layer entity of the gear can be obtained through laser sintering metal powder. 4) Another layer entity can be processed by lowering the workbench to a certain distance, and the complete entity can be obtained by repeating the previous steps. Finally, the entity can be generated by wire-electrode cutting as shown in Fig.18

Envelope tooth surface

Interferenceboundary lineInterference

tooth surface

Transition curve

Fig.18 The interference phenomenon

As can be seen in Fig.18, the minimum tooth width is in the through of pitch curve (tooth labelled-1#). Namely, this position is most likely to interference which verified the correctness of the theory. 8. Conclusion

(1) Combined with the parameters of distribution angle, the tooth distribution regularity is discussed. The tooth distribution of curve-face gear is not in a plane but along pitch curve uniformly.

(2) The tooth width changes with the meshing angle, the minimum tooth width is in the through of pitch curve. Namely, this positions is most likely to interference.

(3) An error evaluation method, based on the extracted points form a series of sections along the tooth length, is put forward. The precision of manufacturing entity and theoretical model can be calculated by solving the normal error of tooth surface.

(4)An approach of the tooth generation of curve-face gear, concerned the tooth profile equation, tooth characteristic analysis, manufacture and error evaluation, is obtained, which can be verified by rolling experiment and additive manufacturing. Acknowledgment

This work was supported by a grant from the National Natural Science Foundation of China (No. 51275537). References Ahamed, N., Pandya, Y. and Parey, A., Spur Gear Tooth Root Crack Detection Using Time Synchronous Averaging

Under Fluctuating Speed ,Measurement, Vol.52, No.2 (2014), pp.1-11.

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© 2016 The Japan Society of Mechanical Engineers[DOI: 10.1299/jamdsm.2016jamdsm00 ]3312

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research on the geometric modeling and tooth-flank

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