Research on calculation of machine-tool settings and flank

Bulletin of the JSME

Journal of Advanced Mechanical Design, Systems, and ManufacturingVol.14, No.4, 2020

© 2020 The Japan Society of Mechanical Engineers[DOI: 10.1299/jamdsm.2020jamdsm0049]Paper No.19-00264

Research on calculation of machine-tool settings and flank modification for spiral bevel gear by duplex spread blade

method based on 4-axis CNC milling machine

Abstract A method is proposed to manufacture spiral bevel gear by duplex spread blade method based on 4-axis CNC milling machine. Firstly, the geometry parameters and machine-tool settings are comprehensively considered. To meet with the tooth taper, a new way is raised to select reference point. Then machine-tool settings are calculated that only first-order mesh performance is guaranteed in the reference point. Therefore conjugate tooth surface is modified by a surface represented as second-order polynomial to optimize mesh performance, as a result a target tooth surface is established, and according to the sensitivity matrix the machine-tool settings are corrected. Finally, the proposed method is illustrated by a numerical example to reveal it’s effective and feasible.

Keywords : Spiral bevel gear, Duplex spread blade method, Reference point, Flank modification, Machine experiment

1. Introduction

Spiral bevel gear is widely used in aviation, aerospace, marine and machine tool for its advantages of driving

smoothly, high transmission efficiency, excellent load capacity, etc. With the improvement of CNC, it is a trend that the machine of spiral bevel gear is converting to completing method from the traditional five-cut method, especially in the developed country of manufacturing. It is more efficient for completing method compared to five-cut method for the machine of pinion is one process for two sides simultaneously by completing, while separated by five-cut method, moreover which is more energy-saving and environmental protection by adopt power dry cutting. Obviously, completing method is an advanced method; however high-end six-axis milling machine is essential, such as Gleason phoenix series and Oerlikon C series.

Duplex helical method is firstly proposed by Gleason to machine spiral bevel gear, while cutting principle and machine setting are not fully revealed (Gleason works, 1971). Tsay et al., (1993) developed a mathematical model can be applied to simulate the tooth surface geometry machined by spread blade and duplex helical method. Gonzalez et al., (2011) presented conversion of the specific machine-tool settings of a given generator to neutral machine-tool settings by duplex heliacal method, and parabolic profile on the blades of the head cutter was applied to adjust the contact pattern. Zhang et al., (2015) revealed the generalized theory of helical method in which gear was non-generated and calculated the basic machine-tool settings by define three reference points, later a mathematical model with circular profile blade was established to adjust contact pattern (2016). Lv et al., (2018) calculated the machine settings of small

Longlong GENG*, Xiaozhong DENG***, Xuemei CAO**, Jing DENG***, Shaowu NIE**and Chuang JIANG***

*School of Mechatronic Engineering, Northwestern Polytechnical University,

127 Youyi West Rd, Xi’an 710072, China

**School of Mechatronics Engineering, Henan University of Science and Technology,

48 Xiyuan Rd, Luoyang 471003, China

***Collaborative Innovation Center of Machinery Equipment Advanced Manufacturing, Henan Province,

48 Xiyuan Rd, Luoyang 471003, China

E-mail: [email protected] Received: 18 May 2019; Revised: 28 October 2019; Accepted: 30 March 2020

1

2© 2020 The Japan Society of Mechanical Engineers[DOI: 10.1299/jamdsm.2020jamdsm0049]

Geng, Deng, Cao, Deng, Nie and Jiang,Journal of Advanced Mechanical Design, Systems, and Manufacturing, Vol.14, No.4 (2020)

module hypoid gear by duplex helical method. All the above research, the calculation of machine settings is complex, and rigorous demands for related machine, such as helical motion, tilt and swivel.

For five-cut method, many exports and scholars have carried out a deep and systematical research. Litvin et al., (1981) suggested the Helixform and Formate methods for Hypoid gears, then realized (1988) the determination of setting of a tilted head-cutter for Hypoid and Spiral Bevel Gears, and proposed (1991) loyal synthesis to calculate machine-tool settings which realize the control of tooth contact and transmission error. Tsuji, I.et al (2011) presented manufacturing method of large-sized spiral bevel gears using multi-tasking machine tool with five axis control. Deng et al., (2014) presented a machining method for face milling using a disk cutter with a concave end by 5-axis CNC machine tool. Yang et al., (2015) proposed a pinion development approach in order to obtain excellent transmission performance of Hypoid and Spiral bevel gears. Jiang (2016) researched the machine setting parameters and cutting test by a new tilt milling machine. Cao (2018) proposed a new method for tooth contact analysis. While for the traditional five-cut method, the machine-tool settings of pinion are calculated based on the machine-tool settings of gear in the reference point by the local conjugate or local synthesis, and the machine settings are different for two sides. And the machine-tool settings are usually calculated in the reference point selected in the middle of tooth surface.

For flank modification, Wang and Fong (2006) proposed a methodology to synthesize the mating flanks of a face-milling pinion with a predetermined fourth-order motion curve. Fan et al., (2008) developed a new approach to correct tooth flank form error by using higher-order coefficients of the universal motions, besides the error surface is represented by polynomials in the paper. Shih (2010) proposed a flank modification methodology based on ease-off topography for face-hobbing. Artoni et al., (2013) presented a methodology to correct tooth deviate that gear deviations mapped into equivalent pinion deviations, and only pinion corrected. Mu and Fang (2017) proposed an ease-off flank modification curvature motion method to solve the problem of the design and fabrication of high contact ratio spiral bevel gear with seventh-order transmission error. Nie et al., (2018) proposed a tooth surface topography modification to improve the meshing performance of spiral bevel gears. And above literatures listed are all about five-cut method or completing, which provided valuable insights for this paper.

To solve the problem of complex calculating machine settings and the constraints of equipment by completing, this paper proposed process spiral bevel gear by duplex spread blade method. The geometry parameters and machine setting parameters are comprehensively considered. And a novel method to calculation reference point is proposed, which is different from traditional method selecting the middle of tooth surface as reference point. Then the geometry parameters are designed and machine-tool settings are calculated in the reference point, and the machine settings are the same to two sides for both gear and pinion. What’s more, the machine settings are without helical motion、tilt swivel. According to the machine-tool settings, ease-off topography is established by the original and conjugate pinion tooth surface to check the original parameters. Only the position of contact determined by machine-tool settings, in order to improve mesh performance, a second-order surface is built to modify the conjugate tooth surface. Finally, the machine setting parameters are corrected that guarantee mesh performance. The logic flow is showed as Fig.1.

Fig.1 The logic flow of design.

2



2. Determination parameters of spiral bevel gear 2.1 Determination geometry and tool parameters

A difference angle of tangent line between midpoints on either side of the tooth is formed comparing single side

method with duplex method is deduced (Zeng 1989).

1 1 2 1 2

0

sin' 1s RP OP P O P

R r

（） (1)

Where, 1 2P P、 represents the points in the middle of tooth flank line, R represents mean cone distance, Orepresents cross point of gear, 'O represents the center of head cutter, s represents the tooth thickness.

Meanwhile, in order to reveal the internal relations between dedendum angle f and 1 , two points are selected, and represented as

2 2tan 2 tan tan cosf

s

L

(2)

Where, represents pressure angle, represents spiral angle, s represents the distance along the tooth thickness, L represents the distance along the pitch line.

It is significant to minimize the difference value between 2 and 1 to avoid excessive tooth taper which has a serious impact on gear strength and tool life. As a result the root line is tilted to obtain a suitable dedendum angle fwhich matches 1 2 , and can be written as according to Eq. (1) and Eq. (2).

0

sintan 12 tan cosf f

s R

R r

（） (3)

Above research is suited for both pinion and gear. Thus it is established to calculate the sum of dedendum angleD as follow equation

1 21 2

0

sin(1 )2 tan cosD f f

s s R

R r

(4)

Where 1f , 2f is dedendum angle of pinion and gear respectively, 1s , 2s is mean normal circular tooth thickness of pinion and gear respectively, and 1s , 2s satisfy the equation 0 1 2( ) 2z s s R , 0z is equivalent number of teeth.

0 0

sin(1 )tan cosD

R

z r

(5)

An assumption is established that sum of dedendum angle s is satisfy with Eq. (5). Then an ideal cutter radius sr can be deduced as

0 tan cossin / 1-180

ss

zr R

（） (6)

Where, s represents the sum of dedendum angle of standard taper. The size of head cutter has been standardized, so a cutter radius r can be selected according to the ideal cutter

radius sr . Once cutter radius r determined, the final sum of dedendum angle D can be calculated based on Eq. (6). Naturally, cut number N can be obtained as

sin20

DN

(7)

3



It is diffebe different f

0N closed to Once the

ta a

Where, The dede

comes true b

1sf

For the rdepth and pre

t2b

h

Where, b

Then the

2.2 Determ The basi

designed. Wia point M is

'sin

rR

It is impof surface.

Pinion adistance. A parameters. T

ferent from thefor couples. Tthe calculated

e cutter numbe

0

6N

ta is profile aendum angle

by tilting root l

010sin 2

sN

root tilted aroesented as

tan f

b is face widthe geometry pa

mination mac

ic parametersith the cut nums selected as t

0 0 tan1n 540 tar z

portant to note

and gear are bsimple modelThe tangent to

e blade that caTo reduce the qd can be selecer 0N determin

angle, upper sif is modified

line about the

ound the midd

h. arameters can

chine setting

(e.g., the toomber changedthe reference p

0

anN

e that the cut n

both generatedl is establisheo the path trac

an be grindedquantity of heted.

ned, the profil

ign is applied to eliminate t mean point. T

dle point, the

be obtained co

g parameter

oth number, td, the mean copoint to calcul

number shoul

d; the only difed as Fig.2,

ce at point M f

Fig.2 In

d in any angleead cutter, a se

le angle ta is c

to inner cuttethe effects cauThe value of m

tooth depth i

ombining that

rs

the pressure aone distance wlate machine s

ld nearby avo

fferent is thata left hand pforms angle

nstallment of h

e; the block is eries of cut nu

alculated as

er; lower sign used by the dimodification c

in heel and to

t of standard t

angle, the spirwill not be satisettings, and d

oiding the refe

t the referencepinion is illus with axis X .

ead cutter.

fixed when pumber is set up

is applied to oifferent of cutcan be repente

oe will change

taper.

ral angle) aresfied with Eq.deduced as

erence point fa

e point is selestrated to calc

produced. Cutp. And then th

outer cutter. t number. Theed as

e, h is the c

e determined . (6). To solve

far away from

ected not at thculate the ma

t number mayhe cut number

(8)

e modification

(9)

change of the

(10)

as the couplee the problem,

(11)

m the midpoint

he mean coneachine setting

y r

n

e

e ,

t

e g

4



Radial d

0rS r

Whereas

arcq

The slidi

bX R

And the

00

m

g

E

X

Then the

3. Establish A unive

coordinate sycradle, (bS xdistance betrepresented aangle ta ,are pinion whichto back 1gX a

distance rS ca

2 20 0( ') 2r R r

s cradle angle

0

0

cosctan' sinr

R r

ing base will b

' tan( )f

other machin

00

e machine sett

hment of dig

ersal model iystem of pinb , , )b bx y z is an

tween cO anas 1q , machinshowed in Figh meets with are equal 0, th

an be obtained

'cosR

q is obtained

be changed as

e setting param

ting parameter

gital tooth su

is establishednion, t( ,t tS x yn auxiliary coond tO , and rne root angle g.3. p represen

1 ap pR .Wey are shown

Fig.3

d as

as

s root line tilte

meters are

rs are calculat

urface

d as Fig.3. S, )tz is coordinordinate systerepresents as

1 , Blank offnts the angle oWhat needs ilin Fig.3, for t

Coordinate sy

ed, and the va

ted.

1 m1 1 1( , ,m m mS x y znate system oem. The mach

1rS , centerffset 1mE and mof cradle durillustration is tthe correction

ystems of mac

riation is dete

) is coordinaof head cuttehine-tool settiroll position

machine centeing the generathat althoughof tooth surfa

chine and hea

rmined as

ate system ofer, c( , ,c cS x y zings, for insta

is the angler to back 1gX

ate, 1 represethe Blank off

ace in the follo

ad cutter.

f machine, )cz is coordina

ance, radial de formed by1 , cutter radiuents corresponfset 1mE and mowing.

(12)

(13)

(14)

(15)

1 1 1 1( , , )S x y z isate system of

distance is they 1rS and m1x ,us cpR , profilending angle of

machine center

s f e , e f r

5



Equation of head cutter in tS can be expressed as

( sin ) cos( sin )sin

r r ( , )cos1

cp p t p

cp p t p

t t p p

p t

R s

R ss

s

(16)

Where, p ps 、 , are parameters of tooth surface, as showed in Fig.3. cpR is cutter radius measured in the plane of 0tz ,and upper sign is applied to outer cutter, lower sign is applied to inner cutter.

Equation of pinion and unit normal vector in 1S can be obtained by coordinate transformation and written as

1( )( )

( )( ) ( )

1 1

1 1

1 p p p t j p t

p p p p p p

p p

1 p p p

p p p p p p

p p

r s , , M , r

r s , , r s , ,

sn s , ,

r s , , r s , ,

s

（）

（）

(17)

Where, 1 ( , )t j pM is transfer matrix from tS to 1S , j represent machine-tool settings. And 1 ( , )t j pM can be detailed description as

11 12 13 14

21 22 23 241

31 32 33 34

440 0 0

t

a a a a

a a a aM

a a a a

a

(18)

11 1

12 1

13 1

14 1 1 1 1 1

21 1 1 1

22 1 1 1

23 1 1

24 1 1 1 1 1 1 1 1 1 1

cos cossin cos

sincos cos( ) sin

cos sin cos sin sincos cos sin sin sincos sin

cos cos sin cos sin( ) sin sin

p

p

r p b

p p

p p

m b r p r

a

a

a

a S q X

a

a

a

a E X S q S

1 1

31 1 1 1

32 1 1 1

33 1 1

34 1 1 1 1 1 1 1 1 1 1 1 1

44

cos( )sin sin cos cos sin

sin cos sin cos sincos cos

sin cos cos sin sin( )) cos sin cos( )1

p

p p

p p

m b r p r p

q

a

a

a

a E X S q S q

a

Meshing equation can be expressed as during the process of generating

( ) 01p p p 1

p

f nr

s , ,

(19)

The theoretical tooth surface geometry can be numerically represented by the coordinates of a series of surface point defined in Fig.4. A grid of n lines and m columns is defined in the axial XL RL plane of pinion that XL is coinciding with axis of pinion.

6



1xXL r

RL r

Where,

The gridchosen accorabout p ps , , for gear. The

4. Flank m

Only theactive designsecond orderThe main ste

(1) Dedu(2) Adju

4.1 Derivat

( ,h h hS x y

systems 1( 1S xsystem (dS xshaft angle foangles that d

1

21

s , ,

s , ,

( )

( )p p p

y p p pr

x1 y1 z1r r r、、 a

d point can berding to Eq. (

p can be obtae models are b

odification oe pressure angn and acquirer surface that eps are as folloucing the conjusting the coef

tion of conju

, )h hz is mesh 1 1, , )1 y z and 2S

, , )d d dx y z is esormed by the etermine the m

21 s , ,(z p p pr

are the three c

e calculated by(11). There arained. Hence, tbuilt of pinion

F

of conjugategle a and spirae a well perfo

tooth lengthwows: ugate tooth su

fficient to mod

ugate tooth s

coordinate sy2 2 2( , , )2x y z arestablished whrotation axes,meshing insta

)

omponents of

Fig.4 Defin

y solving the re three paramthe tooth surfaand gear in U

ig.5 Model

e tooth surfaal angle are ormance, a mwise crowing

urface; dify until a we

surface

ystem which ise rigidly attachose dz is para, E is blank ofantaneous posi

f 1r .

nition of a too

nonlinear equmeters p p ps , , face equation aUG showed in

of gear, pinion

ace guaranteed in

method is put , tooth profile

ell mesh perfo

s rigidly attacched to the piallel to hz . 1 affset the distanition of pinion

oth flank grid.

uation made up in the equatiand unit normFig.5, simulta

n and assemb

n the referencforward to me crowing and

ormance obtain

ched to machininion and geaand 2 are angnce between thn and gear, res

up of Eqs. (19ions. As a res

mal vector are aneously, an a

le.

e point. In ordmodify the cond tooth longit

ned.

ne frame, as sar, respectivelgular velocityhe rotation axspectively.

9)- (20). Refersult, a series oobtained by E

assemble mod

der to realize njugate tooth tudinal twist

shown in Fig.ly. An auxiliay of pinion anxes. 1 、 2 ar

(20)

rence point isof parametersEq. (17). So isdel is set up.

tooth surfacesurface by a

are modified.

6. Coordinateary coordinatend gear. isre the rotating

s s s

e a .

e e s g

7



Similarlysurface param

The gear

d

d

M

L

r

n

Where,

2dM

2dL is theMeshing

( , ,gg

The relat

(21) (d v ω

Where,ω

(2)d

zω

[0d R

12 1m zAccordin

2 (f

y, gear equatimeters of gear

r equation 2 （r

2 2

2 2

d

d

M

L

r

n

2

2

0 sin0 cos1 0

0 0

e 3 3 submatg equation is m

(21)2 ) d d n v

tive speed of m

(2) (1) )d d d ω ω r

(1) [ 1 0d ω

1

2[0 0 1z

z

0]TE ,

2/ z , 1z is the nng to Eqs. (21

( ), gg

ion and unit r.

, ）g g and uni

2

2

cos 0sin 0

0 00 1

trix 2dM . met in contact

) 0

meshing poin

(1)d d d r R ω

0]T ,

1]T ,

number of pin) - (23), 2 ca

normal vecto

Fig.6 M

it normal vect

000

point which e

t can be expre

nion and 2z is than be solved,

or can be rep

Mesh coordina

tor 2 ,g g （）n

expressed in S

essed as

the number of and expressed

presented as 2r

te system.

）can be repre

dS as follow

f gear. d as

2 ,（）g g and n

sented in dS as

2 ,g g （）n , wh

s

here , gg are

(21)

(22)

(23)

(24)

e

)

8



Substitutobtained.

Then the

1

1 1

M

L

r

n

Where

1dM

And, 1dL

4.2 Flank m

A secondaims at get a

1a X

Where, Xcrowing thatprofile crowi

Equationmodification

The quaestablished b

( ,T g r

5. Calculat

The mac

order to obtabroadly deve

First the

ting Eq. (24)

e correspondin

1

1

( )( )

gd d g

d d g g

M

,n

,r

1

1

1 00 cos0 sin c0 0

is the 3 3 sub

modification

d order surfacwell perform

2 22 3X a Y a X

X andY are alt modifies tooing, 3a represenn (26) is appron model can be

antity of modiby Eq. (25) an

1) ( , )gg g r

te correction

chine-tool settain a well m

eloped and hasdeviations be

in Eq. (21) th

ng points in pi

1

1

0 0sin

cos 00 1

E

bmatrix 1dM .

n

ce is developemance which re

XY

long tooth lenoth lengthwisents the factor opriate for bote built; here a

ification is cand Eq. (26), an

1) ( ,g n

n of machine

tings are corrmesh performa

s been provenetween target a

he tooth surfa

inion can be d

ed to modify epresents as

ngth and toothe crowing, 2aof tooth longith convex sidsketch (only

Fig.7 Mod

alculated accond represented

)g

e-tool setting

rected to elimance with the n to be a reliaband original to

ace ,d g g （）r

deduced as

the conjugate

h profile, resp2 represents thitudinal twist

de and concaveconcave side)

dification of to

ording to Eq.d as

gs

minate the devexist gear. T

ble and effectiooth surface c

and vector dn

e tooth surfac

pectively. 1a rehe factor of tothat modifies e side. Adjust) is listed in Fi

ooth surface.

. (26), represe

viations betweThe flank topive method forcan be express

,g g （）of ge

e in order to

epresents the ooth profile crtooth longitud

ting six coeffiig.7.

ented as . T

een target andographical cor reducing mased as follow

ear represente

regulate the c

factor of toorowing that m

udinal twist. icients, and th

The target too

d original tooorrection methanufacturing e

d in dS can be

(25)

curvature that

(26)

th lengthwisemodifies tooth

hen a different

oth surface is

(27)

oth surface inhod has beenerrors.

e

t

e h

t

s

)

n n

9



- ( , )T 1 p pr r r

(28)

Taking machine-tool settings into account, Eq. (28) can be represented as

- ( , , )1 p p j r r r

(29)

Then differential coefficient of Eq. (29) can be expressed

1 1 1 11

1

( ... )p p k

p p k

ss

r r r r

r (30)

Taking dot product of Eq. (30) with the vector 1n

1 1 1 11 1 1

1

( ... )p p k

p p k

ss

r r r r

r n n

(31)

For vectors 1 pr s and 1 pr are perpendicular to the surface normal 1n , so it can be simplified as follows

1 1 1 1 1 11 1 2

1 2

( ... )k

k

r n r n r n

r n (32)

Where 1 r n denotes the deviation of tooth surface, 1 1( ) / jr n denotes the influence coefficient of machine-tool setting.

Using represents the deviations of grid points, and then the sensitivity coefficient of parameter j is as follows

j j

(33)

As mentioned, the tooth surface deviation of grid points which is made up of i ( 2i m n ) points can be written in matrix form as Eq. (34) and simplified as Eq. (35).

1 11 21 31 1 1

2 12 22 32 2 2

2 13 23 33 3 3

1 2 3

...

...

...... ... ... ... ... ... ...

...

k

k

k

i i i i ki k

(34)

( 1, 2,..., 2 ; 1, 2,..., )i ij j i m n j k

(35)

Where, ij represents the sensitivity coefficient of machine setting parameters j in point l , ij represents the sensitivity matrix, i represents the deviations between target and original gird points, j represents the modification of machine-tool settings.

There are far more grid points than the machine-tool settings in Eq. (35), so it is over determined. The least squares method is be used to solve the corrections.

T T1( )ijj ij ij i (36)

The machine-tool settings will be corrected with the corresponding coefficients until the corrected tooth surface have a well mesh performance.

10



6. Numeric6.1 Numeri

A couplemethod. The

The bui

between the

As Fig.8offset referenpoint. For thmore intuitiv

cal example ical example

e that the basimachine-tool

lding of easeoriginal and c

Fig.8

8 shows, for tnce point, thee convex side

ve, a distributi

and machine

ic parameters l setting param

Tab

Parameters

Number of teePitch cone angFace cone angRoot cone angModule/mm Face width/mmShaft angle/° Mean spiral anPressure angleWhole depth/mCone distance

T

ParameteRadial seInitial craMachine Vertical oMachine Sliding bRoll ratioCutter radPoint widCut numb

e-off topograpconjugate toot

(a) Concav

Ease-off topo

the concave sie indifferencee, the biggest ion of deviatio

ne experime

are given in meters are calcble 1 Basic p

eth gle/° gle/° gle/°

m

ngle/° e/° mm e of reference p

Table 2 Mac

ers etting/mm adle angle settiroot angle/°

offset, mm center to backase/mm

o dius/mm dth/mm ber

phy is presentth surface of p

ve side of pinio

ography betwee

ide, the biggee becomes bigindifference ion in tooth su

nt

Table 1 is selculated accordparameters of

point/mm

chine-tool setti

P10

ing/° -652

k/mm

2.1

ted in detail ipinion are esta

on

en original and

est indifferencgger. Then this 0.3125mm urface is draw

lected as a nuding to the prospiral bevel g

Pinion (Left hand)

18 27.21611 31.39276 24.41897

7.5932

13.6108.

ing parameter

Pinion 03.0815 15.27172 624.417

0 0

0.97 186519 1114.3 3.09 12

in the literatuablished, as sh

(b) Convex sid

d conjugate to

ce is 0.30064me contact pattin the root of

wn as Fig.9. Fo

mber exampleoposed methogear.

Gear (Right hand)

35 62.78389 65.58103 58.60724

16 53 90 35 20 6683 6804

rs.

Gear 03.0815

65.27172 58.6

0 0

0.97 .124495 114.3 3.09 12

ure (Nie 2018hown in Fig.8.

de of pinion

ooth surface of

mm in the toptern may be i

f heel. In orderor elastic defo

e to illustrate od, and given i

)

8).The ease-of.

f pinion.

pland of heel,in the aroundr to display co

ormation is 0.

the proposedin Table 2.

ff topography

, and as mored of referenceontact pattern00635mm, so

d

y

e e n o

11



the gradient i

As showroot line andlongitudinal coefficients t

The devi

As Fig.1for the convparameters aalso be correTable4.

is selected to

Fig.9 D

wn in Fig.9, ford there is no e

twist are unthat listed in T

iations betwee

(a

F

0 illustrates, vex side, the bare corrected bected, while

distinguish th

(a) C

Distribution of

r concave anddge contact. B

ndesirability tTable.3 are app

en target and o

a) Concave of

Fig.10 Devia

for the concavbiggest deviatby the methodonly machine

he deviations i

oncave of pini

deviations betw

d convex side But unfortunathat the toothplied to modif

original tooth

pinion

ations between

ve side, the bition is 0.0574

d proposed in e setting para

in different co

ion (b

tween original

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iggest deviati434mm that othis paper. Ne

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ndicular to thewing and tooth

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12

The deviiations betwee

ConcavConvex

Table

Parameradial sinitial cmachinverticalmachinsliding roll rati

en corrected a

Table 3

e of pinionx of pinion

e 4 Correctio

eters setting/ mm cradle angle sene root angle/ °l offset/mm

ne center to bacbase/mm

io

and original to

Adjusting co

1a 0.00027 00.00032 0

on of machine

C-1

etting/ ° 0° -1

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0-0

ooth surface ar

efficients.

2a 0.00018 0.0.00025 -0

e settings for p

Correction 1.240238 .085152 1.525668 .846601 0.012888 .972381 0.015783

re as shown in

3a .00015 0.00013

pinion.

n Fig.11.



As Fig.10.009098mmerrors mainlythe root of toobviously, an

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Referringmore obviouFig.14.

F

1 illustrates, fm that occurs iy exists in the oe, the biggestnd the target toe-off topograof deviations

Fig.12

Fig.13 D

g to Fig.9 andus. Tooth cont

(a) Concave

Fig.11 Deviat

for the concavin the topland edge of the tot error decreaooth surfaces aphy correspos are shown in

(a) Conc

Ease-off topo

(a) C

Distribution of

d Fig.13, the tact analysis i

(a

Fig.14

e of pinion

tions between

ve side, the bid of heel, the ooth surface; se from 0.057are approxim

ond to the con Fig.13.

cave of pinion

ography betwe

Concave of pini

deviations betw

tooth contact is carried out

a) Convex of g

4 Tooth conta

target and corr

iggest error bbiggest error for the convex

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patterns are busing the corr

gear (b

act patterns and

(b) Conv

rected tooth su

etween targetdecrease from

x side, the big009689mm. C

chine setting

(b) Convex o

and conjugate t

b) Convex of p

d and conjugat

becomes big arected machin

) Concave of g

d transmission

vex of pinion

urface of pinion

and correctedm 0.081893mggest error is 0Comparing to

parameters a

of pinion

tooth surface o

pinion

te tooth surface

along the toothne-tool setting

gear

n errors.

n.

d tooth surfacmm to0.009098

0.009689mm Fig.10, the er

are shown in

of pinion.

e of pinion.

th length, and gs, the results

ce of pinion is8mm, and thethat occurs inrrors decrease

n Fig.12, and

the bias in isare shown in

s e n e

d

s n

13



6.2 Machine Based on

pinion and ge

Comparicarried out awithout obvi

In order comparison imade, and th

e experiment

n the approacear are proces

ing the modeafter fillet. Thious vibration.

to make it isis carried out

he position and

ched method,ssed respectiv

(a) G

l (Fig.5) and e scene of rol.

s easy to undthrough the p

d contact path

Fig.17

, an experimeely both two s

Gear process

Fig.15

product (Figlling test and

Fig.16

derstand the mposition of conh are mapped.

7 Real and th

ent is carried sides. Instanta

Machine expe

.15), they arecontact patte

Rolling test an

match of the ntact and the The results ar

heory contact

out by 4-axisaneous states d

(b) Pinion pr

eriments.

e in harmony rns are showe

nd results.

actual tooth ccontact path. re showed in F

pattern comp

s CNC millingduring proces

ocess

with each oted in Fig.16.

contact and thThe rubbings

Fig.17.

arison.

g machine YKsing are show

ther. Finally, The transmiss

he result of cs of real conta

K2260X. Thewn in Fig.15.

rolling test ission is steady

calculation. Aact pattern are

e

s y

A e

14



As showed in Fig.17, (17a) is real contact pattern of gear convex. The angle between contact path and pitch line is 40.5°，and the distance between the middle point of contact and toe is 18.4mm , while the corresponding data showed in Fig.17(c) are 40.9°and 18.96mm. 17(b) is real contact pattern of gear concave. The angle between contact path and pitch line is 40.5°，and the distance between the middle point of contact and toe is 19.72mm,while the corresponding data showed in Fig.17(d) are 41°and 19.63mm.

From the comparison in the position and contact path, results of real contact pattern and TCA are matched. The difference may be caused by machine errors and installment errors during the process.

7. Conclusion

(1) In this paper, duplex spread blade method is proposed to machine spiral bevel gear. Geometry parameters and

machine-tool setting parameters are comprehensively considered. The parameters are deduced in reference point which is calculated by a new way.

(2) A modification by second order surface is proposed to modification tooth lengthwise crowing, tooth profile crowing and tooth longitudinal twist for two sides simultaneously in order to get a well mesh performance.

(3) An experiment is carried out by a 4-axis CNC milling machine, results of real contact pattern and TCA are matched through the comparison in the position and contact path. It demonstrates that the proposed method to process spiral bevel gear by duplex spread blade is effective and feasible.

Acknowledgments

The authors would like to thank the financial aid and support from the National Natural Science Foundation of

China (Grant NO. 51475141, Grant NO. 51675161, Grant NO. 51705134, and Grant NO. 51705135). We are grateful to the reviewers and editors for their valuable comments and suggestions. Reference Artoni, A., Gabiccini, M., Kolivand, M., Ease-off based compensation of tooth surface deviations for spiral bevel and

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Documents

Research on calculation of machine-tool settings and flank