Limiting conditions in gear shaping for corrected involute gears

l n l I M a t h lc,.fl Des R e , Prirllcd iii (He;I[ BlltLiin

qxl2/I " > . 7 , x ~ S ~ 1 ~ - ~1 PctL, atll1,+l~ 1)re,, ] 1,]

\oI 2 ~ N~ 4. pp 227 23-~ 1983

LIMITING CONDITIONS IN GEAR S H A P I N G FOR C O R R E C T E D INVOLUTE G E A R S N. NRINIVASAN* and M. S. SHVNM~_G.~,M, ( Received 13 danua D' 1983: in final form 15 Jura' t 983 )

Al~stract~Thc technique most commonly used to produce cylindrical gear_,,, both external and internal, v, L'ca, shaping with a pinion-type cutter which generates gear flanks with rotating motion and rcmo,.c~, chip,, hx reciprocating action. Though the gear shaping cutter of a spccilic module can bc used to cut eear~, x~ith a wide range ot teeth ot the same module, in practice the inherent errors limit the number of teeth that can be cut on the gear. This paper analyses such inherent errors for both external arid internal gears, taking into account the c[[cct o{ a tip-relief tingle and considering both "'corrected" and "'uncorrected" gear forms.

NOMENCLATURE basic

X,; Z, Z,,

generating pressure angle tip-relief angle of cutter generating centre distance protile shift coefficient of cutter profile shift coefficient o1 gear number of teeth on the cutter number of teeth on the gear

m

module

r,. r,,

tip radius of cutter base circle radius of cutter tip radius of gear base circlc radit,s of gear

0 At X.

r,,:

r,,:

pressure

angle

c(,,

%

1. INTRODUCTION GEAR s h a p e r cutters are the only tools c a p a b l e of g e n e r a t i n g cy l i n d r i cal gears of c l u s t e r , s h o u l d e r , h e r r i n g b o n e and internal types. T h e design a n d m a n u f a c t u r e of these g c a r s h ap er s d e m a n d s the u t m o s t care for satisfactory p e r f o r m a n c e . T h e i n e v i t a b l e need to p r o v i d e a relief angle to the flank s u r f a c e s an d a rake an g l e to the g e a r s h a p e r cutter i n t r o d u c e s errors of form of the i n v o l u t e profile. T o c o m p e n s a t e for these err o r s , profile c o r r e c t i o n has to be i n c o r p o r a t e d in the d e s i g n o f the cu t t er . Fo r this, the cu t te r s have their cutting edges r e l i e v e d with t h e i n t r o d u c t i o n of t i p - r e l i e f angle. A f t e r each r e s h a r p e n i n g of the cutter t e e t h , cu t t i n g e d g e s h a v i n g d i f f e r e n t profile c o r r e c t i o n s are exposed. F o r the present study, pinion cutters with z e r o r a k e a n g l e are c o n s i d e r e d . G e a r shaping introduces i n h e r e n t errors like u n d e r c u t t i n g an d t i p - b e v e l l i n g on e x t e r n a l gears in certain ranges. In internal gears, a p a r t f r o m t i p - b e v e l l i n g of the a b o v e n a t u r e , tip-shearing due to t r o c h o i d a l and relieving i n t e r f e r e n c e s is also p r e s e n t . This p a p e r deals with analysis of i n h e r e n t errors in g e a r s h a p i n g m e t h o d c o n s i d e r i n g both " c o r r e c t e d " and " ' u n c o r r e c t e d " gear forms. T h e l i m i t i n g c o n d i t i o n s are p r e s e n t e d m this p a p e r , which can be used as a guide line to the m a n u f a c t u r e r as well as to the designer.

*Research Scholar. Department of Mechanical Engineering, Indian Institute of Technology, Madra,,-r,(lu (136, India.

+Assistant Professor. Department of Mechanical Engineering. Indian Institute of Technotoe~. Mad~a>-r~!,~ ()36, India. 22-

228

N. SRINP,"~S.\?,, and M.

S. ~tlUNMt'(;XM

2. G E O M E T R Y OF ( ; E A R SHAPING CU'VFER

The sectional view of the pinion type shaping cutter is shown in Fig. 1. Along the design section, the elements of the cutter correspond to those of standard gear and the correction is zero (X, = 0). To relieve the cutting edges and to allow for a n u m b e r of regrindings, the sections above and below the design section are modified. Neglecting the effect of rake angle, it can be seen that the initial section of the cutter can be regarded as a positively corrected gear ( X , is positive). The tooth is thicker in the initial section than in the design section. As the cutter is reground, different sections of the cutter with different profile shift coefficients are exposed, according to X( =

m,

tan 0

(1)

where x is the distance between the secuon under consideration and the design section, and 0 is the tip-relief angle. It should be noted that the tooth profile in each section is composed of the s a m e involute constructed from a base circle of radius, r . , . When the above mentioned cutter is used for cutting gears with a pressure angle c~,,, the generating pressure angle % and the generating centre distance A e will depend on the profile shift coefficient corresponding to the cutting section. This part has been taken into consideration in the following analysis. 3. EXTERNAl. G E A R S H A P I N G

In external gear shaping, the following types of inherent errors are observed: (i) undercutting and (ii) tip-bevelling.

Side relief ~n;I;~

Tip reliefpngtee

x 7i r

/

~,

/

"

/

DesignFisection---i_ naI section?~ a

. . . . . . .

Initial section

.

.

.

//

/

e"

. . I . . .

~/ Rake angle ~t

.

.

'

"..5>

.

i*~

I-

~ oJg

~ec

--i

4=,-

FIG. 1. Section ",'ic~ at pinion ~ypc shaping cutter.

V>z..

--~

//////

-

?"7 5''

__ •,I!2XSm-~.1.25m~ ~-Xcnn

///

///

SectionX-X

Gear Shaping tar ('orrccted Involute (}cars

2>~

Og~ g

Gear A

o c

FIG. 2. U n d e r c u u i n g i n external gear shaping.

3.1. Undercutting U n d e r c u t t i n g occurs w h e n gears with a smaller n u m b e r of teeth are cut by a cutter with a c o m p a r a t i v e l y larger n u m b e r of teeth. U n d e r c u t t i n g occurs when the tip of the cutter projects into the base circle of the gear. T h e limiting condition for u n d e r c u t t i n g to be absent is given as in Fig. 2 [1]. r,,,. ~ O,.A ; . ~< \/,(A~,.sin %)2 + r~,

(2)

where

inv%

A~, =

re(Z(; + Z,) cos oq~ 2 cos %,

r,,, =

m Z, .3

+ 1.25 m + X , m

r( k :

H'I g ~ .~

COS

= inv%

+

Oq~

2(X(; + X,.) ( Z ( ; + Z,) tan %

3.2. Tit?-bevellmg This p h e n o m e n o n occurs when the n u m b e r of teeth on the gear is quite large c o m p a r e d to that of the cutter and is due to the tip of the gear digging into the base circle of the cutter. The condition for tip-bevelling to be absent can be given as in Fig. 3 [11. 161.

r,,,~ ~ OrB tee ~< ~/(A~ sin

%)2 + ~

(3/

2311

N. SRINIVASAN and M. S. SItt:>,Mt ~i.x~.I

Og ~ r

Gear

z

°

,

Gear A

Ag

Cutter Cutter

0c

FiG. 3. Tip-bevelling in external gear shaping.

where r<,~ =

H"I Z ( ; ~-

11l

r°~' =

+ m + X(;.m

ZG 2

cos

ao

.

In case of external gears, for both the errors mentioned above (3.1 and 3.2), using equations (2) and (3), limiting values of Z¢; for various valucs of Z, and for different values of X~. and X c are obtained. The combined results of these two errors are shown in Figs 7-9. 4. I N T E R N A L G E A R S H A P I N G

In the (i) (ii) (iii) (iv)

case of internal gear shaping tip-shearing occurs due to: tip-interference; trochoidal interference; relieving interference: in-feed interference.

Geo~.J

/ ~

'~ A

\

, ,~g~

\

",

eg~

\,\

',kl

~ ~

Cutter

't Pc--7 Og Gear

FiG.4. Tip-interference

in internal

gearshapine.

G e a r Shaping for Corrected Involute (}ear,,

Gear

231

~(inv aec-inv O~g) ' Pc'~

IOc cutter"

_2

eg 6ear FIG. 5. T r o c h o i d a l i n t e r f e r e n c e m i n t e r n a l gear sh:lping

T~C s i n GtC

,

vog)

('nv (][g/!V'~a~~ ---

[

Cutter

OCOu'~er

xl~..L~-Og sin •' Ge0r

) a

Fro. 6. Relieving/feed-ininterferencein interredgearshaping.

4.1. Tip-interference When the tip of the internal gear falls within the base circle of the cutter, the up is bevelled away by the root portion of the cutter [2]. I41 . Geometric conditions for tip interference to be absent are as given in Fig. 4. O~,B

r,.~ ~> .....

F.

reg

. -

-j-

L~

>i ~/tA~, sm %,) + p.~,

where, re(Z(; A,t, ~ ~

r,,~=

inv a~

=

.

.

.

inv

Z,)

cos a.

..

m Zs -2

. "3 .

-

c,. +

COS o' e

m

-

X(;.m

2(X~; (Z(; -

X , ) ';.an o.,, Z. I

(4)

75

150

5

10

~

15

u~-.4,',.

" P/'

'

25 Zc ~

,i, ...

20

5'/ ~ d /4

.~ -"

~ ,4

o o o

~

30

35

40

~ ,

45

XG= 1.00 , Urxder cutting - - - - Tip bevelling

~

50

150 m4

10

.

FIGs 7-9.

5

n9

~

~.j

,, ~

.

20

//

J

/~

25 30 Zc ~

35

/-.0

45

50

,,,,...

55

ill c x l e r n a l gear ~ h a p i n ~

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

"0:025

x~=

Limi[mgcondition~

15

"~

//

.4

XG=O.O Under cutting - - - - - Tip bevelling 75

0

150-

lO

~ ....

5

O0

/~-+025

Xc

15

20

.

25

\

30

\

35 Zc

.

/-0

L.S

I/ r-o.2s,:., ................ //-- o.o

/r----

Xc

50

55

60

65

bevelling

XG=_~. O Under cutting ---Tip

--

70

G e a r S h a p i n g for C o r r e c t c c t l n \ o l u i c

(,e;ir-

23~

4.2. Trochoidal inteCerence This error occurs when the tip of the cutter o v e r t a k e s the tip of the gear during conjugate action. As shown in Fig. 5, the condition to o x e r c o m e this p h e n o m e n a can be

given as in [1], [2]. [4}. O.e - (inv .% - inv . oL,.~,) >i ~t, + (inv ct,,< - in\ (~,,) ZZ,¢ , .

{5)

where

0~, = cos- ~

[4 :+

0, = cos i

tl

,.<.~

2A ~. r,.,

~"

o% = cos t i r"s] "

~ec :

#'dr{

#'lk]

COS

I [ 1",'< .

4.3. Relieving interference During the return stroke of the cutter, the gear is m o v e d in the radial direction to relieve the tip of the cutter. Due to this relieving m o t i o n , the tip of the gear will be sheared and this p h e n o m e n o n is termed as feed-in interference. As shown by Fig. 6, for any position 0<., the tip of the cutter will be tit a distance [ 11,

[21,141. ISlY,

= r,,, sin e'<

where 0',

=

O,

-

(inv

%<

inx %)

-

T h e tip of the cutter will be at a distance g,v = r,,~, sill tt':t where

it' v = O,v 4- (inv %

0 c, = e, .

iilv <,,.~,) Z,

Interference of this kind will be absent, if )'.,>~ )," (i.e.)

r<,~, sin 0'c, >/ r<.< sill 0', Ifor a n \ value el tll-

(f~l

Based on equations (4). (5) and ((9. tile limiting values of Z<; tor tile above m e n t i o n e d errors (4.1, 4.2 and 4.3) are obtained lor different xalues of Z,. ,\, and X<. The combined results of these errors tire shown in Figs lit--12.

b4

t

150

50

~0

__L

0

20

i--I

30

,(,0

I

-0.25

~ ' Xc: :t'"/-L.~.;;, t_.---- 0.0

5O 75 - 0

+025 "-,,

XG = +1"0

Xc.:

200

_}

70

I

Zc ---~,--

5060

I

~:ii~

5

BO

I

l

1

90100110120

I

Tip bevelling . . . . Trochoidol Interference ....... Relieving Interference

.

0 :

f 2

}

0

4--

~

75

15

10

]

//

i

20

~

30

I

LO

i

50

I

r,

....~:.:;. ..

XG = 0"0

Zc~

60

I

70

I

80

I

go

I

I

}001}0120

I

Tip bevelling -- - - T r o c h o i d o l interference .......Relieving interference

~

Finis 11)-12. Limiting conditions in internalgc~Lr shaping.

~

~6c L~ : ~1

~o~ >~ . +

~ .". ......:...:.;-::~

" o~ cs o d~ r~

u

l 0

~

10

'

20

I

..:.'-~

~,o~

,,÷

.

.

I

30

.

6

L

40

I

50

.

~

/ / ......... " . . ...~j~ . . ...~;~. ~:..~z~

.

i

I

70 Zc - - - , . - -

60

;

BO

I

1

i

90100110120

,

....,,

,no,~

~Tip bevelling ---.Trochoid(]l interference ...... Relieving interference

.

.

...,.~~

Lt~ Lt~ 6 c ~ x .+ . , . . . ,,u

÷

XG=-IO

,.,-, u~ 660

!o x :0

o

r5

5C

Gear Shaping for Corrected Involute Gears

235

4.4. ln-/'eed in{erference In i n t e r n a l g e a r s h a p i n g , the c u t t e r is fed-in in the r a d i a l d i r e c t i o n to the a p p r o p r i a t e c e n t r e d i s t a n c e . D u r i n g this i n - f e e d m o t i o n , the tip of the g e a r t o o t h i n t e r f e r i n g with the c u t t e r t o o t h will be s h e a r e d . A n a l y t i c a l t r e a t m e n t of the i n - f e e d a n d r e l i e v i n g i n t e r f e r e n c e s is the s a m e [61 . and is as shown in Fig. 6. To avoid this i n t e r f e r e n c e , the s a m e c o n d i t i o n as a p p l i e d to relieving interference is valid (equation (6)).

5. RESULTS AND DISCUSSIONS B a s e d on the a b o v e analysis, c e r t a i n g u i d e lines a r e f r a m e d up to select the right c u t t e r in o r d e r to e l i m i n a t e or m i n i m i s e t h e s e i n h e r e n t e r r o r s . F i g u r e s 7 - 9 show the limiting c o n d i t i o n s for e x t e r n a l g e a r s h a p i n g . It can be seen that the limiting value of Z¢; i n c r e a s e s as the profile shift coefficient of the gear (X¢;) d e c r e a s e s . A l s o Z¢; is less than X~, (the profile shift coefficient of c u t t e r ) is positive. W h e n X(; is negative it can be n o t e d t h a t the e r r o r s are p r e s e n t e v e n for a new c u t t e r (X, is p o s i t i v e ) . H e n c e , in the case of e x t e r n a l gears, for b o t h p o s i t i v e and zero profile shifts on the g e a r (X¢,), b o t h u n d e r c u t t i n g and t i p - b e v e l l i n g can be a v o i d e d a b o v e a certain value of Z , , for b o t h new a n d w o r n o u t cutters (for X¢ is positive and X, is negative). T h e limiting c o n d i t i o n s for i n t e r n a l g e a r s h a p i n g are p l o t t e d in Figs 10-12. T h e limiting value of Z¢; d e c r e a s e s with Xc; (the profile shift coefficient on the g e a r ) . It can also be o b s e r v e d that the n u m b e r of g e a r t e e t h can be i n c r e a s e d to a v o i d e r r o r s in the case of a new c u t t e r (positive X¢). It is also m o r e for a l a r g e r cutter. F r o m the a b o v e figures, it can be s e e n t h a t in the case of i n t e r n a l g e a r s , all the i n h e r e n t e r r o r s can be e l i m i n a t e d for all v a l u e s o f Xc; a n d X¢., a b o v e a c e r t a i n value of Z,.

6. CONCLUSIONS F o r Z, = 17 and a b o v e , b o t h u n d e r c u t t i n g a n d t i p - b e v e l l i n g of e x t e r n a l gears can be c o m p l e t e l y e l i m i n a t e d in the r a n g e i n v e s t i g a t e d , for a positive profile shift on the g e a r (X(;). T h e limiting value of Z(; is 18. This value i n c r e a s e s as X , (the profile shift coefficient on the cutter) d e c r e a s e s . F u r t h e r i n v e s t i g a t i o n s h o w s that for a negative profile shift on the gear (XG), it is i m p o s s i b l e to e l i m i n a t e b o t h the e r r o r s . T h e r e f o r e . to avoid o r m i n i m i s e these e r r o r s , it is d e s i r a b l e to have Z~ = 19 a n d a b o v e . In the case of internal gears, for p o s i t i v e X(;, the limiting value of Z<; is 70 and Z, is 10. For n e g a t i v e X(;, Z(; is 15, for Z c -- 10. T h e value of Z~; i n c r e a s e s with Z, and d e c r e a s e s as X , (the profile shift coefficient on the c u t t e r ) b e c o m e s n e g a t i v e . T h e difference (Z<; Z, ) is r e d u c e d very much for a n e g a t i v e profile shift on the gear. H e n c e . Z < = 34 can be c h o s e n to a v o i d all the e r r o r s u n d e r e v e r y c o n d i t i o n . T h e critical sides of the curves to be a x o i d c d for e v e r y value of X<; a n d X~. a r e s h o w n s h a d e d in Figs 7-12. T h e r e f o r e it can he c o n c l u d e d that a c u t t e r (old or new) h a v i n g Z , = 34 can bc used to cut b o t h e x t e r n a l and internal gears. In case of e x t e r n a l gears, the profile shift on the g e a r should be z e r o or positive for c o m p l e t e e l i m i n a t i o n of these i n h e r e n t errors. REFERENCES Ill M S. S)lc~r,lti(i,xM, First National Conf. on Mechanisms and Machines. IIT. Bombax (December 19~1) [21 I. S. Pain.. N. R.-XMASW.aM~and M. S. SHUNMUGAM.Proc. ¢~lllAIMTDR ('onL. liT. Bombax ( l(-}TSl. [31 M S. SttUNMI. IGAM, Illl. J. Math. Tool Des. Res. 22, 31-39 1t982). [4] V. F. R..XM,XNOV.RUSS. EngngJ. 42(12)45 (1962). [5] V. P. KolrL'Nn