Approximate Equation for the Addendum Modification Factors for Tooth Gears With Balanced Specific Sliding

~

Mech. Mach. Theory Vol. 31, No. 7, pp. 925--935, 1996

Pergamon 0094-114X(95)00111-5

APPROXIMATE

EQUATION

Copyright © 1996 Elsevier Science:Ltd Printed in Great Britain. All rights reserved 0094-114X/96 $15.00 + 0.00

FOR THE ADDENDUM

MODIFICATION FACTORS FOR TOOTH GEARS WITH BALANCED SPECIFIC SLIDING J. I. PEDRERO and M. ARTI~S Departamento de Mecfinica, E.T.S. Ingenieros lndustriales, U.N.E.D., Apdo. 60149, 28080 Madrid, Spain

(Received 2 May 1994; in revised form 31 July 1995)

Abstract--A simple, analytical method to estimate the addendum modification factors for gears designed to have balanced specific sliding is presented, which is valid for every value of the pressure angle and the addendum. Since the method requires neither iterations nor tabular values, it is extremely efficient for computer applications. Copyright © Elsevier Science Ltd

INTRODUCTION

Tool shift is a widely used technique for gear design which enables one to achieve nonstandard distances between shafts, to improve the loading capacity and to avoid undercut in gears with small number of teeth. Additionally, since the shift has influence on some design parameters (like the gears' safety factors, the expected life, the contact ratio or the size of the gears), it can also be used in order to optimize the design, according to the special requirements of each problem. Some proposals have been made for determining the optimal addendum modification factor according to different requirements. DIN 3992 Standard [!] provides a diagram for the total shift (x + x') and some other ones for sharing it between the pinion (x) and the gear (x'), which give teeth with balanced sliding velocity and stress at the root. DIN 3994 [2] and 3995 [3] Standards recommend x = x ' = 0.5 for increasing the loading capacity. FZG recommendation, cited by Niemann [4], provides formulas for the addendum modification factors for gears with great loading capacity, minimum thickness at the tip and contact ratio greater than 1. Some other proposals, all of them cited by Niemann [4], give the addendum modification factors for equalizing the sliding velocity and the scoring limits of both gears, for equalizing the thickness at the root, for maximizing the loading capacity and for minimizing the thickness at the tip. Other proposals, such as the Belgian one, also cited by Niemann [4], recommend x -- - x ' / > 0 in order not to enlarge the size when the pinion loading capacity is increased. Henriot [5] gave graphic representations of the addendum modification factors required for balanced specific sliding in both gears, which reduces wear and heavy scoring risks. Probably the most widely used nowadays is the last one, though it has some inconveniences. First, graphics are convenient when calculations are done by hand, however, when they are done by computer, graphics should be stored in memory, and consequently they should be scanned and stored, and interpolation routines should be developed. Since values read from a graph are not accurate and interpolations increase the error, obtained results are not very good, so that some approximate equations, as ones in [6], have been developed in order to reduce storage requirements and computation times and to simplify the program logic. Second, Henriot's diagrams give one value for x and x' for each velocity ratio and pinion tooth number, but there are infinite solutions, one for each value of the pinion addendum modification factor. If this solution is not admissible, for example, in the case of pre-established center distance, Henriot values should be corrected. Art6s and Pedrero [7] have developed a computer assisted, graphical method that augments the gear design process by giving some diagrams showing the variation of the design parameters as a function of the initial ones, and carried out a complete study of the influence of the static safety factor, the reliability, the surface's finish, the assembly, the material's properties and the helix angle 925

926

J.I. Pedrero and M. Art,s

on the design. More recently, a new computer program has been developed [8] in order to study the influence of the tool shift on the safety factors and the life of the gears, and consequently to optimize the design from those points of view by means of the selection of the optimal addendum modification factors. For this study, if there are no conditions for x and x', a graphic representation of'their influence on any design parameter, for example, the gears life, could be obtained by means of a diagram in which the life is represented in the y-axis, the pinion addendum modification factor in the x-axis, and the gear one as the parameter of the family of curves. But in order to optimize the tool shifts and other parameter, like the module, simultaneously, a new graphic representation, in which every curve of the family generates a new family of curves, will be required. Since this kind of representation is blurred and not very useful for designers, the condition of equalizing specific sliding in both gears could be imposed in order to get more clear and intuitive representation. However, obtaining the relation between x and x ' from the Henriot's diagram, and correcting them until x takes the value to be represented, is very tedious, and hence an analytic equation is suitable. In this paper, an accurate, approximate equation for the addendum modification factors for gears with balanced specific sliding is presented. By means of it, storage requirements and computation times can be reduced, program logic can be simplified and optimization studies can be carried out.

E Q U A T I O N FOR T H E S P E C I F I C S L I D I N G If curve BP of Fig. 1 is an involute then the length of arc AB is equal to length of segment AP, so that r = rb x/1 + 0 2

(1)

fl = 0 - arctan 0

rb

IO

r

Fig. 1. Geometry of involute.

Approximate equation for the addendum modification factors

927

which are the parametric equations for the involute, where r b is the base radius. Taking into account that dl = x/dr 2 + r z dfl 2

(2)

the distance between the two near points of the curve will be given by r

dl = -- dr

(3)

rb

The radius r' of the point of the gear which meshes with a point of the pinion with radius r, can be obtained from Fig. 2 by using the cosine theorem, resulting r ' 2 = r Z + ( r r + r r ) 2, - - 2 ( r r + r ~ ) ( r r c o s 2 c ~

+ x / r 2 _ _ rr2 COS2@ sin ~b)

(4)

where r r and r~ are the operating pitch radii of the pinion and the gear and ~b the angle between the line of action and the horizontal line (operating pressure angle). This operating pressure angle is not equal to the hob pressure angle, because the center distance for gears produced by a tool shift should be modified as follows (5)

d = do + rnxT

where do is the standard center distance (without shift), m the module and xT the sum of both addendum modification factors, x + x'. Consequently, since the velocity ratio does not change, the operating pitch radius should be rr=~--

l+2~T

(6)

L % r~ e~

r rb

Fig. 2. Meshing radii.

928

J . I . Pedrero and M. Art6s

in which Z is the teeth n u m b e r and ZT the total teeth n u m b e r ( Z + Z ' ) . I f (/)h is the hob pressure angle and rp the reference pitch radius, taking into account that the base radius r b does not change, we obtain ....

rp c o s

=

-

rr

ZT c o s

-

-

-

aT + 2XT COS qSh

(7)

which is equal to cos qSh only if XT is equal to 0. Equation (6) shows that the tooth profile does not depend on the tool shift, because the rack that generates it has a n u m b e r of teeth Z ' = m , so that r~ = rp. The only differences are the a d d e n d u m and d e d e n d u m radii and the tooth thickness, but the equation of the profile is the same in any case. Absolute sliding will be given by 6 = Idl - d/'l

(8)

while the specific one will be equal to 6 divided by the m i n i m u m o f dl and dl'. F r o m equation (3) d ( r 2)

dl -

(9)

2rb

and from equations (3) and (4) d(r'2) = d f f l ) ( 1 2r¢ 2rb \

dl'=

(r, + r~)sin 4~'] ~ /

then

d(r2) 'r + < (

d/-dl'-2cos

r;

(10)

sin_ 1)

(11)

which is greater than 0, and consequently equal to the absolute sliding, when r > rr. Hence, for r > rr, specific sliding would be y

dr- at' - -

r;

= --

rr (r r + r; )sin ~b - x / 7 5 - r 2

dl'

1

(12)

while for r < rr, the same expression for the other gear can be used 7 -

dl' -dl dl

rr -- r; (r r -+- r; )sin ~b - x / r 75 - r~,2

1

(13)

M i n i m u m value of the specific sliding is 2 = 0, which is obtained for r = rr, or what is the same,

r ' = r[. M a x i m u m ones are obtained for m a x i m u m values of the radii, which are the a d d e n d u m ones, r = ra and r ' --ra.' W h e n a tooth is meshing at the tip, another pair of teeth are also in contact, so that load will be shared. The tooth will support all the load at the m o m e n t in which the previous tooth finishes its meshing, but this occurs near the point o f contact of both rolling circumferences, where the specific sliding is very small. Consequently wear failure is more p r o b a b l e at the point at the end o f contact. F r o m Fig. 1 it can be concluded that equations (12) and (13) can be expressed as 2(r>rr)='b 7(r < rr) =

r' rb 0 --1 rb (rb + r~, )tan ~b - rb 0

rb

r~O'

-- 1

r~, (rb + r(, )tan ~b - r(, 0'

(14)

where 0 and 0' are the angle 0 in Fig. 1 corresponding to r and r'. Consequently, the m a x i m u m values of the specific sliding, ~a and 7~, will be those for the a d d e n d u m radii, ra and r ; , corresponding with 0, and 0'a, i.e. *

7a =

rb

rb 0a

r b (r b + r~)tan 4b -- r~Oa

Y~ = r b

r~, 0;

r~,(rb + r(,)tan ~b - r(,O',

1 1 (15)

Approximate equation for the addendum modificationfactors

929

APPROXIMATE EQUATION According to equations (15), the condition for balanced specific sliding in both gears, 7~ = 7a, after some calculations can be expressed as r b r(, r b - r~, - - (16) 0a 0 a tan 4~ where, according to equations (1) and trigonometrical considerations 1

1

~l krbJ 1

1

o:

/(r:yj

q\

- i

l

1

tanq~

~ / ( c - ~ s~ ) 2 --1

(17)

Since the values of the addendum modification factor are usually small, the addendum radius and the operating pressure angle of a nonstandard gear (produced by a tool shift) are similar to those of the correspondent standard gear, without shift. Consequently, the three equations (17) can be approximated to the first two terms of the corresponding Taylor series. So 1

Co+ Cl z

x/z 2 - 1

(18)

where 2z 2 - 1 Co -

( z 0~ _

1)3/2

ZO

Cj - (z~ - 1)3/z

(19)

and consequently, from equations (17), (18) and (19)

O~= Ao + \rbJ =A;+Aj

t

a

tan ~b -- Bo + B~ where A0 and A~ are the values of Co and C1 for Zo = r.o/rb (being rao the addendum radius for the pinion without shift), A; and A; the same for the addendum radius of the gear r;0, and bo and BI those for Zo = l/cos q~h" Taking into account expressions (20), equation (16) may be written as =(rb--r~,) ( cBo+Bi rb(Ao+Ar~) -r~, ( Ao+Ai,r~_~) o]s~)

(21)

Replacing in equation (21) the pressure angle for its expression derived from equation (7), and the addendum radius and the base radius for their expressions ra = m

+a +x

Z rb = m ~ cos ~bh

(22)

930

J . I . Pedrero and M. Art6s

where a is the addendum factor, and doing M = (Aj - Bj ) + (A0 - B0)cos ~bh M ' = (A ~ - Bi ) + (A ~ -- B0 )cos (~h M o = 2a(A~ - A ~ )

ZT

B1 - Al (23)

We obtain MZ

- M'Z"

(24)

+ Mo = Nx + N'x'

which is the approximate equation for the addendum modification factors of gears with balanced specific sliding. Equation (24) gives not the value of the addendum modification factors, x and x', but the relation between them. It means that another condition can be imposed. For example, there could be found shift factors giving specific sliding as small as possible, but in this case, big values for x and x', and consequently big sizes for the gears, will be obtained. Other conditions, like pre-established values for the center distance or the contact ratio, can be also considered. Many authors recommend, for a total number of teeth Z + Z ' t> 60, a total shift equal to 0, in order not to enlarge the size. In this case, as seen in equation (7), operating pressure angle is equal to hob pressure angle, and equation (21) can be simplified as follows

(

rb A o + A 1

--r~, A ~ + A I

1

=(rb--r(,)tanq~h

(25)

Now, doing 1

(cos ~h

K = 2(Al + A ~) \ t a n q~h

K'K0 =

l

(cos ~h

2(Al + A'~) ktan 4~h A 1- A ~ AI+AI

A0 cos 4~h-- A~) A~ cos ~h -- A '1) (26)

a

we get x = KZ - K'Z'-

Ko

(27)

which is the approximate equation for the pinion addendum modification factor, considering the case in which total shift is equal to 0, valid consequently for total number of teeth greater than 60. RESULTS

Addendum modification factors obtained from equation (27) have been compared with those from general equation (16), solved by means of numerical methods. Three groups of gears have been considered: gears with small number of teeth (in which gears of 18, 22 and 30 teeth have been selected), with medium number of teeth (with 45, 60 and 85 teeth), and with great number of teeth (with 150, 275 and 500 teeth), and percentage errors for every combination (except those ones in which the total number of teeth is less than 60), have been calculated, for three values of the hob pressure angle: 20, 25 and 21.17 °, the last one corresponding to the transverse pressure angle of helical gears with normal pressure angle and helix angle both equal to 20 °, giving a total of 120 combinations, all of them with addendum factor a = 1. Results are shown in Table 1.

931

A p p r o x i m a t e e q u a t i o n for the a d d e n d u m modification factors Table 1. Percentage error N u m b e r o f teeth

Percentage error Number of cases

Pinion

Gear

Average

Maximum

Small

Small Medium Great

3 27 27

0.000 1.007 1.197

0.00 3.20 2.18

Medium

Medium Great

18 27

0.048 0.585

0.20 1.43

Great

Great

18

0.057

0.22

120

0.643

3.20

Total

Some examples can illustrate the above. A 30 tooth pinion is to mesh with a 150 tooth gear. Pressure angle is equal to 20 ° and addendum factor is equal to 1. The exact value for the addendum modification factor, obtained from equation (16), is x = 0.2755 for the pinion ( x ' = - 0 . 2 7 5 5 for the gear). Approximate value, given by equation (27), is x = 0.2747. Percentage error is 0.29%. The value provided by the Henriot's graph is x = 0.275. N o w a 30 tooth pinion and a 45 tooth gear will be considered. Exact value for the pinion addendum modification factor is x = 0.1183, while the approximate one is x = 0.1187. Percentage error is 0.37%. In this case Henriot's value is x = 0.124, which is 4.47% greater. But the values for the specific sliding for the pinion and the gear with these tool shifts are 1.3278 and 1.3507 respectively, while those with the approximate ones are 1.3423 and 1.3407, which give better results. Exact tool shift gives a specific sliding for the pinion and the gear equal to 1.3418. Figure 3 gives an intuitive idea about the accuracy of the method. Pinion and gear addendum modification factors have been represented as a function of the pinion tooth number, for several values of the velocity ratio (1.1, 1.2, 1.3, 1.4, 1.5, 2, 2.5, 3, 4, 5 and 10), as in Henriot's graph,

0.5

::::::::::::::::::::::::: .................... 0.4 -,._.

0.3

........ ;;;; .......................................... ::'::::;:':,-,,,...,

........

" " .

,,..

" - ..........................

0.2 ....................................................................................... :::::::::::::::::::::::::..............;;;;;;;;;;;;;;;;?i!!!:;...................

• .,.::::t:: . . . . . . . . . . ...... ,........... _ . . , . ......

"-".--,,.,.-,,.

....... ,,,,

• =::::iiii iiii iii:ii !

0.1

.........

....... ................................ - - " " - -

"----"--.-..,

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

......

Z ~_'S..'.;~-~ Z ~__" .-S_Z : - - -Z.:

0 .... -'

::[]]

,.,

,_

'

...... ....

~).1

.

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

,~.~.

,._:.,..:.:

.........

.

,:=:-.=:

1

. . . . .

; :: [ 1

.

:

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

.v,,.,"i"

4).2 ,