Research on Grinding Dimension of Gear-Shaving Cutter Based on Maclaurin Series

Hongping Yu School of Manufacturing Science

Chengdu University Chengdu, P. R. China hpyu354@cdu.edu.cn

Yilong Gong School of Manufacturing Science

Chengdu University Chengdu, P. R. China

Chengdu Tool Research Institution of China Chengdu, P. R. China

Abstract—Shaving teeth is a common accurate method for cutting gears. The meshing transmission of shaver processing gear is such as a pair of crisscross shaft. Because shaving cutter needs grinding many times on the condition, so that it becomes blunt. How to check and calculate the mesh parameter and these parameters whether meet the choose conditions are important tasks. In addition, these processes are blocking points in manufacture engineering constantly. This paper introduces a rapid look-up table method of calculating parameters based on Maclaurin series and put forward some qualitative conclusions. The experiments results show that the rapidly determine the selected shaving cutter of tooth profile is whether or not within the allowed the workpiece gear-shape error range.

Keywords-gear-shaving cutter; Maclaurin series; checking calculate; grinding parameters

NOTATION

β0 helix angle at reference cylinder

β1 helix angle at reference cylinder

znΣ summation value of cutter’s teeth number and workpiece’s

I. INTRODUCTION

With disc gear shaver cutting gear, it needs to make a series of calibration calculation. Only these conditions of calibration are meet, gear-shaving cutter can correctly processing gear. After the Gear shaver is blunt, it is necessary to grind the cutter along its surface of involutes profile teeth. Outside diameter of the teeth is grinded smaller according to the ground thickness amount. Some questions are raised that: When the normal circular thickness of cutter’s reference circle is grinded Δsn0, how to ascertain the variation Δdao about the diameter of addendum circle of shaving cutter. If the Δsn0 and Δdao is irrelevant, it is necessary to calibrate the parameter data of the cutter which is re-sharpened.

Engineering practice shows that the calculation process about calibration shaving teeth is very heavy and complicated. It’s very trouble that the calculation result will check after a heavy re-sharpened process is completed every time. Using the relationship between cross axis meshing and the differential formula of the check conditions, it could get a group of relation formula which always meet the conditions of gear shaving calibration betweenΔsn0 and Δdao. That is to say: As long as the gear shaver have one-time check before

re-grinding, it is determined the heavy grinding size of the gear shaver don't need to make any calibration calculation and it can manufacture correct processing gear.

II. ANALYSIS OF CALCULATION PRINCIPLE

It introduces the formula of meshing center distance about cylindrical gear pair which use Maclaurin series to calculate fast in the literate [1]. It has a lot of advantages to obtain gear shaving center distance aw0, such as fast speed and high precision, etc. However the most important thing is to know pitch circle diameter and end-face pressure angle including both shaving cutter and workpiece when checking and selecting shaving cutter, it is impossible to promote the follow-up check calculation transaction otherwise. Actually we could figure out pitch circle diameter, αw n , αw t and βw according to introduced fast solution in chapter 1 of literate [1]. Take many factors into account, for example, error accumulation result in loss of data accuracy. Therefore the brief formula of pitch circle diameter could calculate from Maclaurin series directly.

A. Calculation Principle The pitch circle diameter of gear-shaving cutter dw 0 can

deduce from the follow formula:

0 0 0cosw b wtd d (1)

in this formula db 0 is base circle diameter of gear-shaving cutter and αw t 0 is end-face pressure angle. We define the xnΣ is accumulative total of normal modification coefficient between shaving cutter and workpiece.

We can take a derivative to the parameter the xnΣ, and the first-order derivative of d’w 0 is:

00 0 0 0

00 0

’ sec

wtw b wt wt

n

wtw wt

n

dd d tg

dx

dd tg

dx

(2)

it is known from formula (1)-(10) in chapter 3 of literate [1]:

2

0 03 3

1 1 0 0

2 secsec sec

wt n wn w

w wn

d tg ctgz zdx

(3)

In case of xnΣ=0, then

978-1-4673-0788-8/12/$31.00 ©2012 IEEE 1467

0 02secwt n

nn

d ctgzdx

(4)

due to:

3 31 1 0 0sec secn w wz z z (5)

Put (5) and(3)into (2), it is

20 0 0’ 2 sec /w w n wn w nd d tg ctg z (6)

In case of xnΣ=0, then

20 0 02 sec /w w nd d z (7)

Now we take the second-order derivative of d”w0 direct at xnΣ in formula (2) is:

2 20 0 00 0 0 0 0 0 0

2 20 00 0 0

” ’ ( ) ( ) '

[(1 )( ) ( ) '

wt wt wtw w wt w wt w wt

n n n

wt wtw wt wt

n n

d d dd d tg d sec d tg

dx dx dx

d d d tg tg

dx dx

(8) The expression 0( ) 'wt nd dx indicate that it is

second-order derivative of αwt0 direct at xnΣ. therefore we draw a conclusion from (3) that αwn, 0w , 1w is a function of

xnΣ.

In the condition of αw t 0=α t 0 and xnΣ=0 we put (5) and (1) into (8), then deduced that:

2" 2 20 0

0 02

3 2 3 21 1 1 0 0 0

4 sec[ ( 3 )

sec sec 3( )]

w nn

n

dd ctg tg

zz tg z tg

z

(9)

Finally it could take dw0 developed as Maclaurin series under known the dw0’ and d”w0:

2' "

0 0 0 0

( )

2n

w n w w

xd d x d d

At last we obtain the applicable expression to calculate gear-shaving cutter’s pitch circle diameter:

0 0

20

2 20

3 2 3 21 1 1 0 0 0

1 1- ( )( 1.5 )

2 sec3

sec sec2( )

n nw

n n

n

n

x xd d F G E

z z

FG ctg tg

z tg z tgE

z

（ ）

For other notations subscript 1 indicate workpiece’s technical parameter table and subscript 0 indicate gear-shaving cutter’s.

B. Results and Discussions

1) Influence of radial backlash due to reduced tooth thickness of gear-shaving cutter’s grinding.

When normal circular thickness of gear-shaving cutter’s reference circle is grinding byΔsn0 it could lead to lessen centre distance aw0 in shaving process. Consequently gear pair radial backlash become thinner; on the other hand grinding thinner dimensionsΔdao of cutter’s tip circle diameter can result in bigger cutter’s radial backlash. WithΔC meaning cutter’s radial backlash variations, it can draw a formula by series of differential calculus:

0

cos1-sin2

nn ao

wn

C s d

(11)

where, αw n is shaving normal pressure angle;

2) Influence of minimum mesh point’s radius of curvature ρ01 of shaving cutter’s effective tooth profile due to reduced tooth thickness of gear-shaving cutter’s grinding.

When normal circular thickness of gear-shaving cutter’s reference circle is grinding byΔsn0 it could lead to lessen centre distance aw0 in shaving process. Consequently gρ01 become thinner; on the other hand grinding thinner dimensionsΔdao of cutter’s tip circle diameter can result in bigger minimum mesh point’s radius of curvatureρ01. it can draw a conclusion by series of differential calculus similarly:

1 001 02

0 0

cos cos2 sin cos sin

b n an

wn b a

ds

(12)

If making the dimensionρ01 of re-grinding cutter same as before shaving, it should let Δρ01=0. Then deduce below formula from (12):

0 00 02

cos cos sinsin

n b aa n

wn

d s

(13)

or

00 02

0 0

cos sinsin cos

n aa n

wt b

d s (14)

Let this formula is substituted into (11), then:

0 00

cos cos sin12sin sin

n b an

wn wn

C s

(15)

By this formula it is obtained an important conclusion: if the abrasion quantity of gear shaving cutter diameter is according to the above formula, the radial backlash of re-grinding gear shaving cutter increase than before. In other words, checking conditions and process can always ensure accuracy.

III. CONCLUSIONS

Through the above discuss in this paper, it shows that shaving cutter selection must check the following two conditions:

1) There must have sufficient radial backlash between shaving cutter tooth top round and the cut gear root circle;

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2) The involute tooth profile height must meet the

requirements.

At the same time, if tooth tip circle diameter of grinding shaving cutter is in certain range, it is unnecessary to carry out calibration calculation about shaving meshing parameters. However it is greater effect in repairing tooth surface grinding workpiece thickness on tooth profile error, we need to check calculation mesh parameter after grinding, and to change the cutter or scrap it if necessary.

REFERENCES [1] P. T. Tian, etc, Gear Tool Design and Selection Manual. Beijing:

Defense Industry Press, 2010.

[2] H. M. Liu, Tool Design Manual, Beijing: China South Industries Press, 2001.

[3] J. Z. Lu and J. N. Sun, Metal-Cutting Principles and Tool. Beijing: Mechanical Industry Press, 2002.

[4] Sichuan Mechanical Bureau, Complex Tool Design Manual. Chengdu, 1978.

[5] A. K. Das, “Technological heredity in spur gear manufacturing,” Journal of Materials Processing Technology, vol. 91, no. 1-3, pp. 66-74, 1999.

[6] S. M. Che, “The calculation chart of tooth thickness tolerance deviation on Cylindrical involutes Gear,” Journal of Mechanical Transmission, vol. 24, no. 4, pp. 31-34, 2000.

[7] I. Mrkvica, “Finish machining of hardened gears and environmental considerations,” Metal Powder Report, vol. 55, no. 11, pp. 45, 2000.

[8] S. H. Suh, E. S. Lee, and S. Y. Jung, “Error modelling and measurement for the rotary table of five-axis machine tools,” The International Journal of Advanced Manufacturing Technology, vol. 14, no. 9, pp. 656-663, 1998.

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