A form grinding method for manufacturing the variable tooth thickness hob on CNC gear grinding machine

A FORM GRINDING METHOD FOR MANUFACTURING THE VARIABLE

TOOTH THICKNESS HOBON CNC GEAR GRINDING MACHINE

Van-The Tran

Department of Mechanical Engineering, Hung Yen University of Technology and Education, Hung Yen City, Vietnam E-Mail: [email protected]

ABSTRACT

The variable tooth thickness (VTT) hob is usually applied for longitudinal crowning work gear surface with twist free tooth flank. It is an important cutting tool for manufacturing the high precision helical gear. However, the manufacturing process for a VTT hob has not investigated yet. Therefore, in this study, we proposed a method for generating profile of VTT hob by using form grinding wheel on CNC grinding machine. A mathematical model for the tooth profile of VTT hob is established by setting the center distance between the grinding wheel and hob as a second order function of hob’s longitudinal feed movement. A numeral example is presented to illustrate and verify the merits of the proposed form grinding method.

Keywords: variable tooth thickness hob, grinding wheel, CNC grinding machine.

INTRODUCTION

The grinding process is the most popular finishing process because of its high accuracy, efficiency, and flexibility in tooth flank modification. Actually, two major grinding methods with high productivity for grinding cylindrical workpieces are usually used. One is a single-indexing process called form grinding method applying a form wheel(developed by Höfler, Niles, and Gleason-Pfauter) and other one is a continuous-indexing process called generating grinding method applying a worm wheel(developed by Reishauer and Gleason). The form grinding is the most suitable process for achieving flexible profile and lead modifications on the finished tooth flanks. The newly developed gear grinding machines are all Cartesian-type structures with movable axes controlled by computer numerical control (CNC). Because of state-of-the-art CNC technology, these machines can offer precise simultaneous five-axis movement that enables free-form flank modification.

The details the geometric design, tooth surface envelope theory and tooth contact analysis of various gears is published by Litvin and Fuentes [1]. Practical applications of various gears, including gear processing methods, tool design and the strength analysis based on the finite element method. In recently, Hsu and Fong [2] and Hsu and Su [3] proposed a hobbing method for anti-twist tooth flank of work gear by using variable tooth thickness (VTT). Besides, the topologies, contact ellipses and transmission errors of the double-crowned work gear pairs generation by modified hob in gear-hobbing process are investigated. Some related literatures on form grinding process are published. Yoshino et al. [4] proposed a flank correction method for form grinding through compensation of the wheel profile and the position between the wheel and the work gear. Subsequently, in extensive research, Nishida et al. [5] derived the wheel profile using an analytical method. And Kobayashi et al. [6] proposed an optimum contact-line shape by

modulating the setting angle of the wheel for tooth trace modification. Then the accuracy of the gear tooth profile is estimated corresponding to the wheel setting errors [7]. You et al. [8] proposed a model that profiles the CBN shape of the grinding wheels using the geometric properties of the contact points. Radzevich [9] proposed a methodology for calculating the parameters of the plunge shaving cutter and the respective form grinding wheel from measured discrete points of the modified gear. Nishimura et al. [10] offer the ZE series for gear grinding machine that feature higher efficiency, higher precision, lower running cost, and easier operation. In 2009, Chiang et al. [11] proposed a simplified two-dimensional numerical simulation method for form grinding the thread on cylindrical workpieces without solving the simultaneous system equations that produce numerically unstable solutions in the presence of undercutting, interference, or double enveloping. And Chiang and Fong [12] proposed a tilt form grinding method and a methodology to determine the minimum tilt angle for avoiding undercutting and secondary enveloping. Profile and lead modifications are more applied in the past years to fulfill high torque load demands and low running noise [13]. After that, Wei et al. [14] established a mathematical model for calculating the processing error of the rotor tooth profile caused by the form grinding wheel profile. Lately, the free-form flank topographic correction method on a five-axis computer numerical control gear profile grinding machine proposed by Shih and Chen [15, 16].

MATHEMATICAL MODEL OF THE GRINDING WHEEL

Basically, the theoretical axis profile of a grinding wheel is a projection of the involute curve on traverse plane, as shown in Figure-1. Accordingly, the position vector and unit normal vector of the grinding wheel’s right-hand side profile can be expressed in coordinate system S_g( ,x_g y_g,z_g)as follows:

[ ( ), ( ), ( ),1] ,T g  xg  yg  zg 

r (1)

and [ ( ), ( ), ( )] ,

g g g

T g  nx  ny  nz 

n (2)

where

(cos cos sin( )),

g og p p

x r      ₍₃₎

0,

g

y  (4)

(sin cos cos( )),

g og p p

z r      ₍₅₎

sin( ),

g

x p

n    (6)

0, g

y

n  (7)

And cos( ).

h

z p

n    (8)

where is the profile parameters, r_og is the radius of the pitch circle of grinding wheel, andp is the profile angle in transverse section, as shown in Figure-1.

w 

og r



1



2



Figure-1. Surface parameters of the grinding wheel.

Since the grinding wheel surface is a surface of revolution, therefore, the grinding wheel profile is used to generate the grinding wheel surface by rotating the profile defined in Eq. (1) around its axis-symmetriczg. By

( , ) ( ) ( ), w  w  wg w  g 

r M r (9)

and n_w( , _w)L_wg(_w)n_g( ), (10)

The matrix Mwg(w)describes the coordinate transformation from Sg to Sw and is represented as:

cos sin 0 0

sin cos 0 0

( ) .

0 0 1 0

0 0 0 1

w w

w w

wg w

 

 



 

_ 

 

_{ }

 

 

M (11)

and the transformation matrix L_wg(_w) is the sub-matrix ofM_wg(_w)by deleting the last column and row.

After some mathematical operations, the locus of grinding wheel surface can be simplified as follows:

( , ) [ ( , ), ( , ), ( , ),1] ,T w  w  xw w yw  w zw  w

r (12)

and ( , ) [ _w( , ), _w( , ), _w( , ),1] ,T w  w  nx  w ny  w nz  w

n (13)

where

(cos cos sin( ))cos ,

w og p p w

x r       ₍₁₄₎

( cos cos sin( ))sin ,

w og p p w

y r        (15)

(sin cos cos( )),

w og p p

z r      ₍₁₆₎

sin( )cos ,

w

x p w

n     (17)

( ) sin

sin ,

w p w

y

n      (18)

and cos( ).

w

z p

n    (19)

By using Eqs. (12) and (13), the 3D model of the grinding wheel surface is shown in Fig. 2.

Figure-2. 3D model of the grinding wheel surface.

MATHEMATICAL MODEL OF THE VTT HOB

a. Mathematical model of the theoretical VTT hob

The profile of the generating surface of the standard rack cutter is the same as that of the standard helical gear. And the theoretical VTT rack cutter profile is generated by modifying the rack cutter tooth thickness of the standard rack cutter as a second order function of the lead parameter, _. 2

t t

s b v , as shown in Figure-3(a). Accordingly, the position vector and unit normal vector of the right-hand side of the theoretical VTT rack cutter profile can be expressed in coordinate system ( , , )S x y zt t t t as follows:

2

. ,

[ , , ,1] [ cos , sin t, ,1]

T T

t x y zt t t  ut on ut onb v vt

r ₍₂₀₎

and [ , , ]T [sin ,cos , . cos ] ,T t nxt nyt nzt  on on b vt on

n (21)

where u1 and v1 are the surface parameters, on is the

pressure angle of the standard rack cutter andbis the VTT coefficient of rack cutter, as shown in Figure-3(a).

According to Figure-3(b), by applying the homogenous coordinate transformation matrix equation, the locus and unit normal vectors of VTT rack cutter surface represented in coordinate system Sht(xht,yht,zht) are attained by

1 1

1 1

1 1

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

( , , ) ( ) ( , ) ( , ),

0 sin cos 0

0 0 0 1

t o t o t og t t t

t o t o t og t t t

ht t t t ht t t t t t t t

o o

r r

u v u v u v

       

       

 

 

  

 

 _{ } 

 

  _{ }



 

 

 

r M r r

and n_ht( , , )u v_{t t} _t L_ht( )_t n_t( , ),u v_{t t} (23)

where L_ht( )_t is the upper-left (3 3) sub-matrix of the (4 4) homogeneous coordinate transformation matrix

( ) htt

M .

According to the theory of gearing, the equation of meshing between the right-hand side rack cutter surface and left-hand side hob surface can be obtained by

[ ( , , ), ( , , ), ( , , )] ( , , ) ( , , ) ht t t t ht t t t ht t t t 0

ht t t t ht t t t

t

x u v y u v z u v

f u v  u v    

 

  



n (24)

t

u

t

v

t

O yt

t

z

t

x

2

.

t t

s b v

on 

t  1

y

ht

y

og

r

2

y

2

O

3,t

O

.

t og t

l r 

t

y

ht

x

,1 ht

z

2

z

1,2

x

3

z zt 3,t

x

1 o 

3

y

,1 ht

O

(a) (b)

Figure-3. Surface parameter of the VTT rack cutter and coordinate systems for the schematic generation of the theoretical VTT hob.

b. Mathematical model of the proposed VTT hob surface generated by form grinding wheel

For generating the profile of VTT hob, a 3D model of CNC form profile grinding machine is presented in Figure-4. This model has six-axis movements that need for grinding profile of VTT hob: rotational axis A₁ is applied for setting the crossed angle between the grinding wheel and hob axes, rotational axes B1 and B2are applied

for controlling rotation angles of the grinding wheel and hob, respectively, horizontal axisX₁ is applied for controlling the radial feed along the center distance between the grinding wheel and hob, vertical axis Y1 is

applied for setting the position of the grinding wheel and hob, axisZ2 is applied for controlling the longitudinal

feed along hob axis. The coordinate systems for grinding process of hob are shown in Figure-5, wherein coordinate systems Sw( ,xw yw,zw) and Sh( ,xh yh,zh) are rigidly

connected to the grinding wheel and hob, respectively, while the coordinate system S xc( ,c y zc, )c is rigidly

connected to the frame of grinding machine, and ( , , )

a a a a

S x y z ,Sb( ,xb yb,zb) andSd( ,xd yd,zd)are

auxiliary coordinate systems for simplification of coordinate transformation. The crossed angle hw of the hob and work gear axes is usually a machine-tool setting. To generate the profile of VTT hob, this study proposes a grinding method by changing the center distance between the grinding wheel and hob as a second order function of hob’s longitudinal feed movement in grinding process. The center distance between the grinding wheel and hob is proposed as follows:

2_.

t o c h

E E a l (25)

2

Z

1

X

1

A

1

B

2

B

1

Y

Figure-4. 3D model of CNC form profile grinding machine.

h W

g W

t E

h 

, ,

c b a

x

b

y za w,

b

z

c

y yd

d

x

h

x

h

y

,

h d

z

w

y

h

l , , b a w

O

,

d h

O

c

O

w

x

a

y



t

E

hw 

Figure-5. Coordinate systems for the form profile grinding machine.

The position vector r_w( , _w)(see Eq. (12)) and unit normal nw( , w)(see Eq. (13)) present locus of grinding wheel surface, respectively. By applying the homogeneous coordinate transformation matrix equation from Sw _to Sh, the locus and unit normal vector of grinding wheel surface can be represented in coordinate system Sh as follows:

( , , , ) ( , ) ( , ),

h  w h lh  hw h lh  w  w

r M r (26)

and nh( ,  w, , )h lh Lhw( , )h lh nw( , w), (27)

where

cos cos sin sin sin cos

sin cos cos sin cos sin

( , ) .

0 sin cos

0 0 0 1

h hw h hw h t h

h hw h hw h t h

hw h h

hw hw h

E E l

l

     

     



 

 

_{ } 

 



   

 

 

M (28)

and the transformation matrix L_hw( , )_h l_h is the sub-matrix of Mhw( , )h lh by deleting the last column and

row.

( , , , ) [ ( , , , ), ( , , , ), ( , , , ),1] ,T h  w h lh  xh   w h lh yh   w h lh zh  w h lh

r (29)

and ( , , , ) [ _h( , , , ), _h( , , , , _h( , , , ),1] ,T h  w h lh  nx   w h lh ny   w h l nh z   w h lh

n (30)

where

) (

( t w cos wsin wcos ) sin ,

h h hw hw h

x  E x   z  y   (31)

) )

( wcos wsin cos ( t w sin ,

h hw hw h h

y  y  z    E x  (32)

cos sin ,

h h w hw w hw

z  l z  y  (33)

)

cos ( sin cos sin ,

h w w w

x x h z h w y hw h

n n   n  n   (34)

)

( sin cos cos sin ,

h w w w

y z h w y hw h x h

n  n  n   n  (35)

and cos sin .

h w w

z z hw y hw

n n





In the gear grinding process, there are two independent kinematic parameters lh and hthat need for

finish tooth flank of work gear. Therefore, there are two equations of meshing between the hob and the grinding wheel as follows:

1

( , , , ), ( , , , ), ( , , , )

( , , , ) h w h h h w h h h w h h 0,

w h h h

h

x l y l z l

f    l         



_ _

  



n (37)

and ₂( , , , ) h( , w, , ),h h h( , w, , ), ( ,h h h w, , )h h 0,

w h h h

h

x l y l z l

f l

l

        

     



n (38)

The tooth surface of the hob can be defined by solving Equations (29), (30), (37) and (38), simultaneously.

NUMERICAL EXAMPLE AND DISCUSSIONS

The purpose of this example is to verify the merits of proposed form grinding method. The basic data of the hob and the form grinding wheel are given in Table-1. The grinding machine setup data are calculated according to the basic meshing conditions as illustrated in Ref. [1].

Table-1. Basic data of the hob and the form grinding wheel.

Items Value (Unit)

Number teeth of hob (N1) 1

Normal module (m_pn) ₂ Normal circular-tooth thickness of

hob (spn1) 3.141 mm

Normal pressure angle (pn) 20.0

Helix angle of hob (p1) 87.8R.H.

Hob pitch radius(rp1) 26.05 mm

Hob form radius(rf1) 24.21 mm

Hob outer radius(ra₁) 28.05 mm Hob length (Fw₁) 135 mm Grinding wheel radius(rog) 41.0 mm Center distance between grinding

wheel and hob (Eo) 87.05 mm Crossed angle between grinding

wheel and hob(hw) 87.8 

Center distance variation

coefficient (a) 1.34×10

-5

VTT coefficient (b) 1.0×10-8

Figure-6. Topography of the theoretical VTT hob surface (maximum deviation of the tooth flank

d

M = 10.0m and ratio of deviationR_d 0.67).

Figure-7. Topography of the proposed VTT hob surface generated by form grinding wheel (maximum deviation of the tooth flankMd 10.23m and ratio of deviationRd 0.66).

CONCLUSIONS

In this study, a new form grinding method for manufacturing the VTT hob is proposed by setting the center distance between the grinding wheel and hob as a second order function of hob’s feed movement (Eq. (25)). Mathematical models for the tooth profile of the theoretical VTT hob and proposed VTT hob are established. The compared results show that the proposed VTT hob surface generated by form grinding wheel fits well with the theoretical VTT hob surface. The proposed VTT hob surface is much smoother than the theoretical VTT hob surface.

REFERENCES

[1] F. L. Litvin, A. Fuentes. 2004. Gear Geometry and Applied Theory. 2nd _{ed., Cambridge University Press,}

Cambridge, UK.

[2] R.H. Hsu, Z.H. Fong. 2011. Novel Variable-tooth-thickness Hob for Longitudinal Crowning in the Gear-hobbing Process. Thirteenth World Congress in Mechanism and Machine Science, Gearing and transmissions, Guanajuato, Mexico.

Top land

Gear root 19.97

24.46 29.97

22.20

27.19 26.71

24.52

22.25 unit: m

23.40 25.63 26.71

25.63 24.52 23.40 22.25

29.97 27.19 24.46 22.20 19.97 Gear root

Top land

Gear root 30.18

24.43 20.0

28.93

22.17 28.23

24.42

24.35 unit: m

26.69 28.72 28.23

28.72 24.42 26.69 24.35

[3] R.H. Hsu, H.H. Su. 2014. Tooth Contact Analysis for Helical Gear Pairs Generated by a Modified Hob with Variable Tooth Thickness. Mechanism and Machine Theory. (71): 40-51.

[4] H. Yoshino, K. Ikeno. 1991. Error Compensation for Form Grinding of Gears. Transactions of the Japan Society of Mechanical Engineers Part C. 57(543): 3652-3655.

[5] N. Nishida, Y. Kobayashi, Y. Ougiya, N. Tsukamoto. 1992. Profile Calculation Method for Form Grinding Wheels (1st report). Grinding Wheels for Helical Gears and Avoidance of Interferences. Journal of the Japan Society for Precision Engineering. 58(4): 628-634.

[6] Y. Kobayashi, N. Nishida, Y. Ougiya and H. Nagata. 1995. Tooth Trace Modification Processing of Helix Gear by Form Grinding Method. Trans. Jpn. Soc. Mech. Eng., Ser. C. 61(590): 4088-4093.

[7] Y. Kobayashi, N. Nishida, Y. Ougiya and H. Nagata. 1997. Estimation of Grinding Wheel Setting Error in Helical Gear Processing by Form Grinding. Trans. Jpn. Soc. Mech. Eng., Ser. C. 63(612): 2852-2858.

[8] H.Y. You, P.Q. Ye, J.S. Wang, X.Y. Deng. 2003. Design and application of CBN shape grinding wheel for gears. International Journal of Machine Tools and Manufacture. (43): 1269-1277.

[9] S.P. Radzevich. 2005. Computation of parameters of a form grinding wheel for grinding of shaving cutter for plunge shaving of topologically modified involute pinion. Journal of Manufacturing Science and Engineering, Transactions of the ASME. 127(4): 819-828.

[10]Y. Nishimura, Y. Ashizawa, T. Katsuma, Y. Yanase, K. Masuo. 2008. Gear Grinding Processing Developed for High-Precision Gear Manufacturing. Mitsubishi Heavy Industries Technical Review. 45(3): 52-56.

[11]C.J. Chiang, Z.H. Fong, J.T. Tseng. 2009. Computerized simulation of thread form grinding process. Mech Mach Theory. 44(4): 685-696.

[12]C.J. Chiang, Z.H. Fong. 2009. Undercutting and interference for thread form grinding with a tilt angle. Mech Mach Theory. 44(11): 2066-2078.

[13]A. Türich. 2010. Producing profile and lead modifications in threaded wheel and profile grinding. Gear Technology 27 (1): 54-62.

[14]J. Wei, Q. Zhang, Z.Z. Xu and S. K. Lyu. 2010. Study on Precision Grinding of Screw Rotors Using CBN Wheel. Int. J. Precis. Eng. Manuf. 11(5): 651-658.

[15]Y.P. Shih, S.D. Chen. 2012. A flank correction methodology for a five-axis CNC gear profile. Mechanism and Machine Theory. (47): 31-45.