Specifying Surface Roughness for Spur and Helical Gears

Tribology for Energy Conservation / D. Dowson et al. (Editors) 1998 Elsevier Science B.V.

SPECIFYING

SURFACE

ROUGHNESS

FOR SPUR AND HELICAL

267

GEARS

W Scott and DJ Hargreaves School of Mechanical, Manufacturing and Medical Engineering, QUT, GPO Box 2434, Brisbane Australia 400!

ABSTRACT This paper looks at the rouglmess profiles of gear tooth surfaces and atten~ts to put the subject of surface topography into context with lubrigzation theory and so help the designer in s p e c i ~ g the optimum finish for the gear teeth. 1 INTRODUCTION

The microgeometry of most surfaces of engineering interest is very complex and it is not possible to uniquely describe a surface by a single number. Nevertheless, a single value such as root-mean-square height deviation (Rq) is often all that is given to specify surface topography. In some cases (including gears) no values are given and the surface microgeometry is inherent in the specified finishing process, eg hobbed, ground, planed, etc. It is now generally agreed however that at least two parameters are necessary to give a good indication of the properties of surfaces. One of these parameters relates to asperity heights and the other to asperity spacing (or line density). Although the terms texture, roughness and finish, in relation to surface topography, are sometimes used synonymously there is a distinct difference in their meaning The current Australian Standard 25361982(1) gives the following definitions: Surface texture - the topography of a surface consisting of both the roughness and the waviness but excluding errors of form. Surface roughness - the topography of a surface which consists of short wavelengths only. It comprises surface irregularities with relatively small spacings and usually incorporates irregularities resulting from the method of manufacture. Waviness - the topography of a surface which consists of wavelengths of intermediate length. Waviness may result from such factors as machine or work deflection, vibration and chatter or heat treatment.

These terms are used for the quantitative assessment of surface "finish" which may be described by the finishing process. Intuitively one would expect the microtopography of the tooth flank surfaces to have some influence on gear durability. This has been proven to be the case by Wellauer and Holloway (2). However, the exact nature of the surface interactions is yet to be determined. Here we will consider the specification, production and measurement of surface topography and show why it is important to the reliability of lubricated gearsets. The most common method of measuring surface roughness is by profilometer whereby a lightly loaded sharp diamond stylus is traced across the surface and its movement convened to an electrical analogue of the surface profile. The signal so obtained is analysed to describe the surface profile by statistical values. In general, surfaces will contain irregularities with a large range of spacings. Profilometers are designed to respond only to irregularity spacings less than a given value. This is called the cut-off. The effect of reducing the cut-off is to suppress more of the longer wavelengths as illustrated in Fig 1. It can be seen that the signal being processed is substantially different in each case highlighting the importance of specifying the cut-off when taking a measurement. When s p e c ~ g cut-off the designer should ensure that the production process will not result in wavelengths greater than the cut-off, or that, if these wavelengths do occur, they will not be detrimental to the functioning of the surface.

268 |.~

.1..mm

A

(a) Measurod Profile Without Electrical Filtering

Co) With 0.8ram Cut-off- 3.5 to 4.2 )am RI

(¢) With 0.25ram Cut.-.off- 1.8 to 2.2 lam R 1

(d) With 0.08 mm Cut-off- 0.95 m 1.05 lam R 1 NOTE: Profiles are distorted by unequal vertical vs horizontal magnification Fig 1 Effects of Various Cut-offs (I) A further complication due to scale effects has been illustrated by Williamson and Majumdar (3). They show that the mean slope and curvature of a rough surface depend on the resolution of the measuring instnnnem. Fig 2 gives the dependence of the ratio of root-mean-square (rms) values at a magnification of 13with that at magnification of unity, on the magnification of the surface. The measurements for 13<40 were obtained by optical intert~rometry and those for 13>40 by atomic force microscopy. It is clear that although the rms height does not depend on the instrument resolution, the rms slope and curvature (which are related to wavelength) show a strong dependence. 2 NOTATION major axis of ellipse, L minor axis of ellipse, L semi contact width, L

pinion diameter, L gear wheel diameter, L Youngs modulus of elasticity, FL"2 equivalent modulus of elasticity, FL"2 elliptical integral of the second kind E hardness, FL"2 H lubricant film thickness, L h ratio of major to minor axes a/b k face width of gears, L L mean radius of asperities, L r pitch radius, L R RI equivalent radius at the contact, L levelling depth (height of the maximum profile Rp peak above the reference line), L average levelling depth, L Rpm The reciprocal of the curvature sum in the x Rs and y directions, L maximum peak-to-valley height, L Rt Rtm average peak-to-valley height, L

Dp DG E El

269

root-mean-squared roughness, L Equivalent radius of curvature in the xprincipal plane, L equivalent radius of curvatt~e m the yprincipal plane, L load per unit width, FL"! load, F pressure viscosity coefficient, L2Fl viscosity at atmospheric pressure, FL'2T ratio of film thickness to composite surface roughness h/o Poisson's ratio composite surface roughness, L root-mean-square value of peak height distribution, L normal pressure angle, deg plasticity index helix angle, deg Subscripts 1,2 - for pinion and gear respectively

Rq Rx

W

W Ot

rio

V 0 O*

Instrument 101

4

M." I 0 ' Jr 10 ~

[.

................... ..,o'

m ....

_

....

Re~)lutlon ; .,o'

[jam]

. ........... ...~o'

cu,v**u,e. ,.

~

III rms x-cucvat~e

/

.

.

1

.... Q

: . ,o" ,

--

I0"

~i t0 j t

!0:

" 0

~

o" Io'

Thus the real area of contact is the sum of these microeontact areas which is usually very much less than the area associated with the geometric boundaries, normally referred to as the apparent area of contact. Greenwood and Williamson (4) have shown that three topographical properties are important in static contact. These are density of the asperities on the surface, the standard devia!ion of their heights and the mean radius of their stnnmits, all of which are included m their plasticity index. This index, which gives an indication of the mode of surface deformation ie elastic or plastie, is defined as

K=H'" N R

(1)

The larger the plasticity index, the more critical are the mterfacial conditions. If r>l a large part of the contact is plastic, for r-z:).6 the chances of plastic flow are remote and most of the contacting areas will be subject to elastic deformation. Although this model is oversimplified for gear tooth contacts it is useful in determining the influence of surface topography parameters on the interfacial conditions for the boundary lubrication mode. However, as shown by Fig 2 the values of K will be instrument resolution dependent.

I~ I0°1

I i0 °

I0'

I0" Magnification.

,8

i0 ~

tO"

Fig 2 Variation of rms height, slope, and curvature as a function of instrument resolution (3) 3 GEAR LUBRICATION

Gears normally operate under lubricated conditions either in the boundary or elastohydrodynamic (EIqD) modes. The application of lubrication theory to gears has reached the stage where surface roughness is being taken into account in an effort to select the optimum lubricant. When two surfaces are in contact under boundary lubrication conditions they touch at the peaks of their microgeometric irregularities.

Although the boundary mode may occur m gear lubrication, the elastohydrodynamic 031-1D) mode is more prevalent. In EHD lubrication a lambda 0-) factor, defined as the ratio of the calculated lubricant film thickness between the teeth and the composite root-meansquare (o) rouglmess, of the surfaces has been introduced as an indicator of the expected surface durability of the teeth. McCool (5) has extended Greenwood & Williamson's model and applied it to rolling contact bearings under lubricated conditions with varying lambda (~) values. It is equally applicable to gear tooth contacts.

270

-i.+-++++.

h

p.

!00

N/ram m Q N/ram I 0 290 N/~ A

:

"

t8~

4.

2a

I.

Tootrt II Circ, speed : ! 0-~ T o o t h s u r f a c e "|R t : 3.7~m : 0.7 ~lm I

2b

IRa

I Oil

0.0+

0.0

"~ . . . . . . . . .

/

Inlet

................... -

.z

2.80

role

hobbed

temo.:

-

CLP

•

.s

150

"

-

/

20"C

• .....................

.1

"

(iJml

t.0

:

I00 185

tO l.;t

....

I

i

Average Oil Film Thickness

(Note: The vertical scale in these i!lusttationl i s ma(nified much In~e than II~e horizontal $¢11l. Real surfaces 4o net ha+e such hil~ S$op~l.~

).,l!

Fig 3 Schematic of Film Geometry (5) It is possible to have two film surface geometries having the same mean separation but with quite different surface roughnesses as shown m Fig 3.

~,.0-

1.0-

For the situation in Fig 3a, the probability of surface contact would be small, that is a lubricant film would always separate the metal surfaces. In Fig 3b, there is a high probability of surface contact. Therefore it is important to know the surface roughnesses as well as the average film thickness when describing the lubricant film geometry. Theoretical work by Patir and Cheng (6) and others, and experimental work by Peeken et al (7) have shown that the surface texture affects the calculated film thickness. This is attributed to the gap geometry influencing the retention of the lubricant in the contact area. Fig 4 from Peeken et al (7) quantifies this effect for different gear tooth finishes. The k, parameter relates the lubricant film thickness, h (based on "smooth" surfaces) and the composite surface roughness, o of the bearing sul-faces.

k : h/o

where o = (Rql 2 + Rq22) 'a

(2)

PH

Nfmm z Nlmm z

0

Tooth ! I 2 9 0 N/mm I A Clrc, sD~ea " 2 . 8 0 m / s T o o t h s u r f a c e hol~beO anO p o l t s n e 0 Rz • 0.7 vm R+ • 0 , 2 pm 011 / inlet te~." CCO 150 / 2 0 " C

o.o .+ ........ : . ............ .

.

.s

-: ......-_.

.7 [pml

~.o

I

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

Fig 4 Lubricant film thickness m film contact (7) Rql, Rq2 are the root mean square roughnesses of the bearing surface,s. The specific film thickness therefore gives an indication of the thickness of the lubricant film compared to the roughnesses of the surfaces in contact. It has been shown that a value of 3 or 4 indicates that there will not be any surface contact. Lower values of ~ say 1.5 indicate that there is considerable interaction between opposing surface asperities, that is penetration of the lubricant film. One aspect which has not been taken into consideration is how the spacing or wavelength of the asperities affects both o and h when detennimng suitable values for these parameters. It is necessary therefore to compare asperity wavelength with the apparent contact area of meshing geartecth.

m

271

4 APPARENT CONTACT AREA OF MESHING GEAR TEETH Researchers in EHD theory have consistently reported that the contact patch for EHD is very n ay the same as that calculated for dry Hertzian contact. This section uses Hertzian theory to evaluate the contact areas in spur and helical gear sets.

When spur gears mesh the apparent contact area will be a long thin rectangle assuming perfect alignment. The dry Hertzian contact semi-contact width can be found from equation (3) taken from Wirier and Cheng (8).

{w,;/,°

(3)

Calculations for a pair of slow speed heavily loaded gears with a pinion pitch circle diameter of 0.48m showed that the contact width (2b) was about 3.Smm on the pitch line. 4.2 Helical Gears

The situation for helical gears is far more complex than that for spur gears. The contact shape is an ellipse and this contact ellipse moves along the tooth face as meshing takes place. The meshing action can be represented by vimml spur gears in the plane normal to the tooth surface (Merritt (9)). By so doing and uti!ising the simplified technique developed by Brewe and Hamrock (10) and outlined by Winer and Cheng (8), the dimensions of the contact ellipse can be estimatecL Rx = Rl sin o

cos 2 ,

Ry =

R 2 sin o cos 2

k = a/b = 1.0339 (Ry/Rx)0636

a=

(4)

(5)

(6)

k 2EwRs) t/3

-~.

E

0.5968 E = 1.0003 + Ry/Rx 1

Rs

4.1 Spur gears

b = 1.598

6

E=2

--

I

Rx

El

+

1

......

Ry

+

l

.

(7)

(8)

(9)

(lo)

Note that there is no significant loss of accmacy when using this technique for I sRy/Rx s35.5, that is for le; k <10. Using the above relationships the contact dimensions for several gear sets are given in Table 1.

$ GEAR TOOTH SURFACE MEASUREMENT Measurements of the tooth surfaces of "as machined" gears were taken using different parameters and cut-offs. As an example, the rouglmess profile of an AGMA Class 10 pinion tooth, taken in the radial direction, is illustrated. The profile trace is shown in Fig 5 where it can be seen that wavelengths comparable with cut-off (the shaded bar) are present The numerical results for three different cutoffs are given in Table 11 We have found that, in general, the values of roughness which we have measured are much higher than those quoted as being typical, see Table 111.

272

Table I

(a) Spur Gears ~

=

_

:::::

:

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

.

. . . . .

Do(m)

Dp(m) 0.068

.

.

.

0.250 .

.

.

.

.

.

.

,.

.

.

.

.

.

.

.

.

.

.

Power (kW) .

1

. - ,

.

.

.

.

- , , . ,

Gear Wheel Speed

.

.

I

48 _

2

0.126

0.171

3

0.128

0.168

26

4

0.126

0.186

-

800

21

.

0.186

"

.

'

.

.

~ .

.

.

.

.

.

-

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

.

.

.

6

0.100

0.150

330

7

0.96

0.96

220

.,,

.

.

.

.

0.157

.

.

.

.

.

.

.

.

.

0.295 .

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

=

._

.

0.776

.

2.75

.

.

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

.. . . . . .

24 .

.

"

.

0.826

.

.

.

0.431 ..

2000 .

.

.

0.386

1800 .

(mm)

.

1800 _

0.126

L

Contact Width

1000

....

5

.

1720

. °

0~M)

.

.

.

(b) Helical Gears ,,

,.,,

. = ,

_

,u,,u, . . . .

.J._

Do(m)

Dp(m)

|

Power

Gear Wheel Speed

Contact Dimensions (mm)

(I~M)

0~vO

.

1

0.162

1.258

1119

578.00

0.293

1.987 .

0.556

.

.

.

1119 .

.

.

.

.

.

.

2.3 90

.

.

.

26.5 °

6.00

6.11 °

1900

..

,

,,

,,,,,

Ra

Rq

Rt

R3z

Rsk

5.03

6.29

43.2

21.8

0.33

2.95

0.II

13.4

14.5

9.5

0.24

Average Cut off 0.25 .....

,.

~ ,

..j,,,~

,L

....

,.:,.,,WUL..

""""""It...='

.

. '

. . . . . .

'

. . . . . . . .

5.738 x 1.505 .

.

.

.

..,

.

,

.,

..........

_

.=

_

_

10.524 x 3.015 31.040 x 14.590 .

.

.

.

.

.

.

.

.

.

_ _ . . : _

~ .

Initial Value lam lain

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

Average Cut off 0.8 2.34

.

:-:--

.....

- .

~

...........

_ . . - -

.

- .

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

Typical Values of Composite Roughness, (Mobil (5))

Tooth Finish

Average Cut off 2.5 26.0

.

Table m

Measured roughness in radial direction (pro)

4.42

.

,

Table ll

3.36

.

.

. . . . .

m t , . _ . _ . w

.

.

85.30

. . . . . . . .

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

.

28.35 ° l

2

.

Hobbed Shaved Lapped Ground- Soft - Hard Polished

1.78 1.27 0.33 0.89 0.51 0.18

70 50 13 35 20 7

~,,.au

Run-in Value lain lain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.02 1.02 O. 19 -

.,,.~

40 40 7.5

-

........

~

.__

273

Rottghness Profile :

. . . . .

".-

$1'3+ v 6 :: -

Rank Taglor Hobson

"

" : ........

"" ..............

'......."'- ..................

t

~, t

--

~-

-"11

I III

I

I

.

.

.

.

.

.

.

.

.

.

I'

J. i

ttI

E

.

Ra um 3.36 Bq um 4.42 Rt u~ 26,0 R3z uM 13,4

\

Rsk ~x.., "%.,,.,

Cutoff

8. II

8.8

Roughness I$0

. . . . . . .

--=-=

. . . . . .

-

. . . .

-.

~

. . . . . . . . . . .

Bear, in9 Rat, io

.. . . . .

...: ....................

P!etric Uv 3841

AmpI i t u d e dens i t ~

Fig 5 Roughness Profile of AGMA Class 10 Pinion Tooth (the shaded bar indicates the 0.8mm cut-offvalue)

6 DISCUSSION

From Table II it can be seen that the roughness values for the surface profile given in Fig 5 differ with the cut-off value. This means that there are wavelengths greater than 0.8ram and less than 2.5mm. Since the calculated contact width for this meshing pair was 2.75mm, the wavelengths between 0.8 - 2.5mm will influence the contact gap geometry and thereby affect the film thickness, h, and the specitic film thickness, k. Hence, for this application the 2.5mm cut-offwould be more appropriate. On the other hand for spur gears 2-5 in Table I a 0.8mm cut-off may be the most appropriate as the contact width is less than 0.8mm However, for gear 1 a cut-off of 0.25mm may be best for the 0.157mm contact width.

Roughness m the axial direction will also affect the lubricant retention and this should also be measured and compared with the contact dimensions. This implies that, for spur gears, the signal of the roughness profile should be unfiltered (ie the cut-off would be the traverse length of the instrument). For helical gears the cut-off would be chosen in conjunction with the major and minor axes of the contact ellipse which we have found to have an extremely large range for different applications (Table I). The Australian Standard (1) states that in the absence of a specified cut-off, a value of 0.8mm shall be assumed and the surface roughness evaluated at this cut-off However, the results reported here for gear contact, width (for spur gears) and ellipse (for helical gears), would suggest that in many cases a

274

measurement based on a 0.8mm cut-off would not include all of the information which would affect the lubricated contact and that the concept of functional filtering should be adopted.

2 Wellauer, EJ and Holloway, GA, Application of EHD oil film theory to industrial gear drives, Trans ASME, d Eng Ind, 98B(2), May 1976, 626634

7 CONCLUSIONS

3 Williamson, M and Majumdar, A, Effect of Surface Deformations on Contact Conductance, Trans of the ASME, Vol 11, November 1992, pp 802-810

EHD lubrication theory is being used by some designers to predict the performance of gears with regard to tooth surface durability. Since this theory involves the tooth surface texture then this property should be specified before manufacturing, and measured thereafter. In many cases this does not occur and the resultant surface texture is inherent m the finishing process. The ~, value used to determine the likelihood of gear tooth surface distress takes account of roughness heights only. It would appear that roughness spacing is equally important. Measurements which we have carried out by profilometry give results which indicate that the chosen instrument cut-off is an important and often neglected value which should be selected with reference to the Hertzian contact geometry. The Rq roughnesses which we obtained by measurement were consistently greater than those quoted in the literature as typical. Since it is clear that asperity spacing affects both the roughness and film thickness values, more work is required to determine the relationship between tooth surface texture, so measured, and the durability of the gears under both boundary and EHD lubrication conditions. 8 REFERENCES

1 AS2536-1982, Surface texture, Association of Australia, Sydney

Standards

4 Greenwood, JA, and Williamson, JBP, Contact of nominally flat surfaces, Proc Roy Soc, A295, 300, 1966 5 McCool, J'l, A computer program for evaluating surface roughness, Ball Bearing Jnl, 234, Sept 1989 6 Patir, N and Cheng, HS, Application of Average Flow Model to Lubrication Between Rough Sliding Surfaces, ASME d Lub Tech, 101, 220 (1979) 7 Peeken, H, Ayanoglu, P, Knoll, G and Welsch, G, Industrial Oil Performance in Terms of Base Oil Chemical Structure, Lubrication Science, Vol 3, No I, October 1990, pp33-42 8 Winer, WO and Cheng, HS, Film thickness contact stress and surface temperatures, Wear Control Handbook 9 Merritt, HE, Gear Engineering, John Wiley and Sons, New York, 1971 10 Brewe, DE and Hamrock, B J, Simplified solution of elliptical contact deformation between two elastic solids, Tram ASME, J Lubn Tech Series F, Vol 99 No 4, October 1977