Analysis of Traction Forces in a... Kannel 1986

7/25/2019 Analysis of Traction Forces in a... Kannel 1986

1/7

J. W . Kanne l

Battelle Columbus Laboratory,

Columbus, Ohio 43201

T. A. Dow

Precision Engineering Laboratory,

North Carolina State University,

Raleigh, N.C. 27695-79 10

Analysis of Traction Forces in a

Precision Traction Drive

A theory for the shear stress between a rough elastic cylinder and a cylinder w ith a

soft laye r has been developed. The theory is based on a Fourier tran sform approach

for the elasticity equations coupled with surface deflection equations for transient

contacts. For thick layers (h > .001 in.) the shear stress on the surface approaches

the shear of the layer alone. The elastic shear deflection (-100 \xin.) as a result of

the tangential load is significant and increases if a surface layer such as a thin

coating is added to one or both cylinders. The predicted interfacial shear stresses are

considerably altered by surface roughn ess on uncoated surfaces and these effects

are ameliorated by the addition of a thin soft surface coating.

Intr oduc t ion

Two crit ical aspects of a precision machine are l inear

location of one part relative to another and smooth motion

between lim its. One m eth od of achieving this result [1, 2] is to

use a traction drive on the slideway as i l lustrated in Fig. 1.

Positioning accuracy below the microinch level is typically

required. W hen accu racies of this level are involved, virtually

all factors which affect motio n must be considered in order to

minimize errors in the system. One such factor is the shear

deformation of the drive system, especially the elasticity of

the traction interface. The elasticity is affected by many

factors, including the Young's modulus of the traction

components, surface layers on the rollers (such as solid fi lm

layers) and the roughn ess of the rollers and slideway.

The most extensive work reported on the analysis of the

traction interface is by Kalker [ 3 , 4 ] . Kalker traces the traction

interface between two extremes: the Cattaneo [5] problem and

the Carter [6] problem. The Cattaneo problem occurs when a

cylinder is rotated slightly, while in contact with a stationary

surface. The Carter problem occurs when both the cylinder

and the mating surface are moving but at sl ightly different

speeds. Kalker 's study traces the traction forces through the

transients between the two extremes.

Bentall and Johnson [7] analyzed the slip between two

dissimilar cylinders in rolling contact. This research allowed

for tangential deflections due to microslip. Barber [8] con

ducted research similar to Kalker 's, only he analyzed three-

dimensional contacts of rollers under misalignment. Poritsky

[9] derived basic equations for cylinders in contact and

discussed the problem of rough surfaces. Krause and Senuma

[10] did experimental studies with rollers which developed

surface corrugations. The surface corrugations notably af

fected the tractio n beh avior of the cylinders.

The work presented here is an extension of the Kalker and

Poritsky work with allowances for surface layers. The surface

layer algorithm is developed from the work of Sneddon [11],

and Gupta and Walowit [12].

Contributed by the Tribology Division of THE

AM ERICAN SOCIETY

OF

MECHANICAL ENGINEERS and presented at the A SM E/A SL E Joint Tr ibology

Conference, A tlanta, Ga ., October 8-10, 1985. M anuscript received by the

Tribology Division, A pril 19, 1985. Paper N o. 85-Trib-45.

Leaf spring

Hydrostatic bearings

Drive motor

Traction bar

Idler roller

Capstan

Hydrostatic

bearings

The resolution of the slide drive will be

0.2pin. (0.005yumj. This is the equivalent to

0.05 arc.

Fig. 1 Illustration of traction control system for precision engineering

A p p r o a c h

The same general approach used for the normal stress

analysis [13] can be used for the shear stress computa tions. A s

will be shown subsequently, the matrix equation is almost

identical to that developed by Gupta-Walowit for the normal

stress computations. That is the deformation equations can be

put in the matrix form:

CF=e

(1)

where C is the matrix equation with elements c,y that relate the

tangential de flection, e, to the tangential forces. A s with the

normal stress equations, a relationship between point-loads

and po int stresses can be developed.

Matrix Coeff ic ients

For a solid body, Po ritsky derived a similar relationship for

load-d eflec tion coefficients as for nor ma l stress coefficients,

that is

4(1

v

2

)

7rE

In I/-1

(2)

Journal of Tr ibology JULY 1986, Vol . 108/4 03Copyright 1986 by ASME

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1

-s

2

1

- z

Z( v

x

' -

_

(1 + " '

where

z-

? =

1)

)

3 - z -

(1 +

1-

=lsl

=

exp(z)

0

I

\-z

\'(\-z)

)-2z/s

2

1

s

1

7

2

zt

1

( l - v , ' ) z ?

2

(1 + I V ) ?

2

v'=(2-v)/(\-v)

v

=v/(\-

V

)

Table 1

0

s

-,2

( 1 + z ) ?

2

[2+ ( l + z ) ( l - V ) J '

2

[(l + v

x

) + 2z/s

2

]l

2

0

0

- ?

z?

( 1 - K

2

' ) Z ? ( y

2

'

- ( 1 - K

2

" ) 7 ? . 2z7? /s'

3)7?

s

2

c,

Di

A

2

B

J

--

-i

=

=

1

0

0

0

^oj

/3= ( 1 - ^

2

) E , / ( 1 - . ,

2

) E

3

7 = ( l -

J

' 2

2

) E i / ( l - " i

2

) E

2

wherer is the distance between a point tangential load and the

corresp onding tange ntial deflections. A s with the norm al

stress equations, a relationship between point loads and

stresses over a small region is:

F:

rdx

(3)

where Tis the surface shear stress acting over a distance dx (or

A x in finite term inology). Then

-4(1

-v

2

)

7rE

j -

\n\x:x\dx (4)

Inlet

F ig .2 S l ippage in the contac t zone

which allows a solution to be found whenx

;

= x

h

A n expression has been developed for the influence-

coefficient matrix for shear stresses in a layered solid (see

A ppendix A ) . The so lu t ion (assuming no normal pressure on

the surface) can be written as:

7rE

[ J o

2 5

=i

where e,(0) is the assumed initial deflection relative to a fixed

poin t (x

0

= - 106). For subsequent t imes equation (7) is used

with AV At being a constant that is added to each time step to

produce a given traction.

In the computations the coefficients c,y were set and the

shear stress computed by a matrix solution of equation (12).

A t some poin ts

Tj >f'Pj

wh ere / is a coefficient of friction an dP j is the local pressure

computed using the technique given in reference [12 and 13].

For these points the matrix was adjusted as follows:

n

Y, l--8jj

)(c

u

-c

0J

)TjAx=e

i

/ =

n

-H h

f

fPj^iJ-

c

oj)-r

Jf

=fPj

f

(13)

where

I

1

J=Jj

j

are the po in ts where r

y

>

fpj.

The essential size of the

matrix will be reduced by one row for each value ofjf. The

computations involved a simple i teration starting with a full

matrix and subsequently reducing the matirx for each value of

Tj > fPj until further i terations produced no changes.

D isc uss ion

Figure 3 i l lustrates the shear stress at the interface for th e

case of a stationary lower cylinder (Vt/b = 0) and for a series

of relative slip values

(Vt/b

> 0). The slip values are ex

pressed in terms of the half width, b. I f Vt/b = 0.8, a point

on the upper cylinder has moved a distance equivalent to 80

percent of the half-width of con tact between the cylinders.

For the case of a stationary lower disk (analogous to the

Cattaneo problem), the shear stresses are the lowest in the

center of contact and rise toward the edges. This rise is due to

the contribution of each element to support the shear

deformation outside the contact region. The rise in shear

stress is limited at

x/b =

0.6 by slip between the cylinder

surfaces; tha t is, the shear stress becomes equal to the friction

coefficient t imes the normal stress. Thus, the shear stress

curve has the same shape as the normal stress distribution

where slip is present. For this example the horizontal load was

105 N /m m (600 lb / in . ) , and to produ ce th is force, the upper

cylinder was rotated 2 X

10

~

4

rad (the upper surface moved 2

/mi).

Cont inued ro tat ion of the upper cy l inder would produce

rotation of the lower disk because the loading was assumed to

remain constant (analogous to Carter problem). Different

values of the upper cylinder motion are also i l lustrated in Fig.

3 . For the largest rotation of the upper cylinder shown in Fig.

3 (Vt/b = 1.2, correspon ding to 3.2 x 10

2

rad (about 2 deg)

of rotation), a point on the upper cylinder has moved a total

of 2.5 ^m from a point on the lower cylinder which was

adjacent at no load. This means that the driving cylinder will

rota te 2.7 x 1 0

4

rad more than the driven cylinder for an

average rotation of 3.2 X 10 ~

2

ra d against a load of 6001b.

One purpose of the analytical traction model was to

Journal of Tr ibology JULY 1986, Vol . 1 08/40 5



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-0. 8 -0.6 -0.4 -0. 2 0 0.2 0.4 0.6 0.8

Distance From Contact Center, x/ b

Fig .

4 Con tact zone shear stress distribution (0.01 in. layer)

b = .89 mm (.035 in.) Load = 543 N/mm (3099 lb/in.)

F, = 14.5 N/mm (83 lb/in.) R=9.44 m m (.37 in.)

E T =2 GPa

290 ksi)

p

h

=.44 GPa

64

ksi)

e

= 2.5 pm (100 ^in.)

evalaute the role of surface layers on traction drive per

forma nce. Figures 4 to 6 show the shear stress distribution for

the condition of a uniform tangential displacement of 2.5 xim

but with a soft surface layer (E = 2 GP a) on one of the

cylinders. Surface layer thicknesses of 250 /xm, 25 ttm and 2.5

/xm are illustrated.

A ll curves show peak shear stresses at the edges of contac t

and reasonably uniform shear stress in the center. For the

thicker layers this center shear stress should approach the

shearing of a soft layer of known thickness a known amount.

That is:

T

G

m

y G

m

e/h (14)

where

7 is the shear strain

G, is the shear modulus of the layer (G, = E /2 ( l + v) =

7 80 M Pa)

For the 250 /xm layer shown in Fig. 4, the predicted shear

stress using equation 14 is 7.8 M Pa (1.125 ksi). This predicted

stress is consistent with the stress near the center of contact of

Fig. 4. A t the edges of the contact (x = b) the stress rises

considerably above this level to compensate for the forces

required to tange ntially deflect the layer outside of the contac t

region. Very near the edges, the shear stress is l imited by the

coefficient of friction times the normal pressure.

In Fig. 5 the predicted shear stress using equation (14)

would be 78 M Pa (11.25 ksi). This stress is higher than the

stress at the center of contact given in Fig. 5. That is, for thin

films, the shear stresses tend to be high enough to deflect the

substrate as well as the surface. When the substrate is

deflected this deflection must be subtracted from

e

in com

puting the surface shear. For very thin fi lms (h = 2.5 /xm) as

shown in Fig. 6 the center shear stress is considerably lower

than predicted by shearing of the layer alone. For this case the

equatio n (14) shear stress would be 780 M Pa (112.5 ksi) or

about four t imes that given in Fig. 6. Clearly then the

evaluation of this surface films requires the use of com

prehensive theories and cannot be achieved by simple

analyses.

Figures 7 and 8 i l lustrate the compariso n between a layered

and a nonlayered body. For the conditions given here a

surface "wind-up" of 2.5 / t in produces a tangential load of

105 N /m m w hen the soft layer (E = 2 GP a) is in place. If

there were no layer, 105 N /m m could be obtained with a

- 0 .8 - 0 .6 - 0 .4 - 0 .2 0 0 .2 0 .4 0 .6 0 .8


Fig . 5 Con tact zone shear stress distribution (1000 in. layer)

b

= .44 mm (.0173 in.) Load = 490 N/mm (2800 lb/in.)

F

(

= 45 N/mm (258 lb/in.)

R =

9.4(.37 in.)

E = 2 GPa 290 ksi) p

h

=.8 GPa 117ksi)

e= 2 . 5 / i m ( 1 0 0 x 1 0 % i n . )

45

40

35

p

s

" o

30

S

t

e

S 20

CO

15

10

5

-

~ / ^\

I I I

-

y \ -

\ -

i i i

- I .0 -0 .8 -0.6 -0. 4 -0. 2 0 0.2 0.4 0.6 0.8 I .O


Fig . 6 Con tact zone shear stress distribution (100 p in. layer)


F

(

= 103 N/mm

591

lb/in.) R= 9.4 mm (.37 in.)

E=2G Pa 290ksi) p

h

= 1.28 GPa 187ksi)

c

= 2 .5 / im(100/i in.)

"wind-up" of 2/xm, as indicated in Fig. 8. The presence of the

soft layer also increases the amount of "wind-up" required to

move the driven cylinder against the load. For example, for

the bare cylinder (Fig. 8) a " w in d- up " of 2.5 /xm occurs in

traversing through 1.2 half widths 0.3 mm). For the layered

cylinder (Fig. 3) a "wind-up" of 3.5 tun occurs in traversing

the same distance.

For the above examples the surface of the drive cylinder

would move 2.5 t tm farther than the driven cylinder over 0.3

mm traverse. If one of the cylinders contained a soft layer the

differential tra verse w ould be increased by ab out 1 /xm. In a

precision control system, all errors must be minimized to

reduce the level of error compensation required of the control

system. Based on error minimization alone then, i t would

seem that bare cylinders would be superior to coated cylin

ders. However, when surface roughness factors are included

40 6/V ol . 108, JULY 1986 Transact ions of the ASME



5/7

-1.0 -0 8 -0, 6 -0.4 -0. 2 0 0.2 0.4 0.6 0.8 10

Distance From Contact Center, x/b

Fig.

7 Conta ct zone shear stress distribution (100p in. layer)


F

t

=105 N/mm (600 lb/in.) R=9.4 mm (.37 in.)

D

=

2

G Pa

290

ksi)

p

h

=

1.28 GPa

187

ksi)

-1.0 -0.B -0.6 -0.4 -0. 2 0 0.2 0.4 0.6 0.8 I.0


Fig.

9 Shear stress distribution with 1 p in. CLA surface roughness (no

layer)

b =.25mm (.01 in.)

F, = 105 N/mm (600 lb/in.)

E = 200GPa(29Mp s i )

= 1.9 pm (75 pin.)

Load:

R-

Pma x

=

:639 N/mm (3649 lb/in.)

9.4 mm (.37 in.)

2 GPa (295 ksi)

-I.O -0.8 -0.6 -0. 4 -0.2 0 0.2 0.4 0.6

Distonce From Contact Center, x/b

Fig .8 Conta ct zone shear stress distribution (no layer)

b

= .25 mm (.01 in.) Load = 607 N/mm (3470 lb/in.)

F, = 105 N/mm (600 lb/in.) R=9.4 mm (.37 in.)

=

200

G Pa

29M

psi)

p

h

= 1.5 GPa

219

ksi)

in the shear stress examinations the value of a surface layer

becomes clear. Figure 9 indicates the shape of the shear stress

distribution for a stationary lower cylinder without a layer but

with a 0.025

pm

center line average (cla) surface roughness.

The increased deformations due to the surface roughness

produces peaks in the shear stress distribution in the slip

regions

(x/b >

0.7). The addition of a soft, thin layer (2.5

nm )cushions the surface asperities and produces a smoother

shear stress curve as indicated in Fig. 10. Based on the results

of Figs. 9 and 10, it would be difficult to compensate for the

erratic stresses for a nonlayered cylinder. However, it is quite

reasonable to attempt to predict the stresses where a layer is

present.

- 10 - OB - 0 6 - 0 .4 - 0 .2 0 0 .2 0 .4 0 6

OB

10


Fig . 10 Shear stress distribution with 1 p in. CLA surface roughness

(100 pi n. layer)

b = .

29 mm (.0115 in.)

F

(

= 100 N/mm 601lb/in.)

=

2

G Pa

29

ksi)

e= 2.5pm 100pm)

Conclusions

Load = 545 N/mm (3114 lb/in.)

R = 9.4mm(.37 in.)

Pmax =

1

-

3 3 G P a

Traction drive systems represent reasonable devices for

traversing slideways in precision machining. However, one

inherent problem with traction is that the traction interface

must incur sizable elastic "wind-up" before the driven

cylinder will move against a given load. For one specific case

analyzed the "wind-up" was on the order of 2 ^m for a 105

N /mm traction load. The amount of "w ind-u p" increased as

the driven cylinder moved.

In order to achieve precision control in a traction drive,

some type of compensation algorithm must be employed to

eliminate "wind-up" errors. It would be expected that a

compensation algorithm of the type presented herein could be

employed provided good reproducibility of the traction in-

Journal of Tr ibology JULY1986, Vo l . 108 /407



6/7

te r face could b e achieved. O n ep r o b l e m in r e p r oduc i b i l it y is

surface roughness . Even smal l levels o f roughness (0 .025 t im

cla)

c a n

cause wi ld var ia t ion s

in

in ter fac ia l shear s t resses .

A

th in coat ing such as a 2 .5fim mo lyb den um disul f ide coat ing

c a n a bs o r b t h e r oughne s s a n d crea te a muc h s moo t he r ( a n d

hence more reproducible) shear s t ress d is t r ibut ion . T h e

pr e s e nc e o f t h e soft layer wou ld cau se a slight (0. 6 i tm)

increase

i n

" w i n d - u p "

b u t

p r e s uma b l y

t h e

" w i n d - u p " w o u l d

be predic table .

.Body 3

,v

. 4 $ 2 b *

T ( X )

) h Body 1

t

w

7

y

r

E..

'

Body 2

References

1 Bryan, J . B., "De sign and Construction of an Ultraprecision 84 Inch Dia

mond Turn ing M achine,"

Precision Engineering,

Vol. 1, N o. 1, 1979, pp.

13-17.

2 Barkman, W. E. , "M achine and Tool Drive Sys tem,"

Precision Engineer

ing,

Vol. 2, N o. 3, 1980, pp. 141-146.

3 Kalker, J . J . , "Trans ien t Ro l l ing Contact Phenomen a,"

Trans. ASLE,

Vol. 14, 1971, pp. 177-184.

4 K alker, J . J ., "R oll ing With Slip and Spin in the Presence of Dry Fric

t i o n , "

Wear,

9, 1966, pp. 20-38.

5 Cattan eo, C , "Sul Contatto di du Corpi Elast ici: Destribuzione Locale

Degli Sfoizi ," fiend. Acad.

Lincei,

Series 6, Vol. 27, 1938, pp. 342-348,

434-436, 474-478.

6 C ar ter , F . W., "O n the A ct ion of a Locomot ive Driving Whe el ,"

Proc.

Royal Soc, a 112, 1926, pp . 151-157.

7 B entall , R . H., and John son, K. L., "Slip in the Roll ing Contact of Two

Dissimilar Elast ic R ollers ,"

J. Mech. Eng. Sci.,

Vol. 9, 1967, pp. 389-404.

8 Barber, J . R., " The R oll ing Contact of M isaligned Elast ic Cylin ders, " / .

Mech.

Eng. Sci.,

(I . M ech. E.), Vol. 22, N o. 3, 1980, pp.. 125-128.

9 Poritsky, H., "Stress and Deflections of Cylindrical Bodies in Contact

Wi th Appl icat ion to Contact Gears and L ocomot ive Wheels ," A SM E

Journal

of Applied Mechanics, 1950, pp . 191-201.

10 Krause, H., and Senuma, T., " Investigation of the Influence of Dynamic

Forces on the Tribological Behavior of Bodies in R oll ing/Sliding Contac t W ith

Par t icu lar Regard to Surface Corrugat ions ," A SM E

JOURNAL

OF

LUBRICATION

TECHNOLOGY Vol. 103, 1981.

11 Sneddon, I . N . , Fourier Transforms, M cGraw-Hill , 1951.

12 Gupta, P . K., and Walowit , J . A ., "C ontac t Stresses Between a Cylinder

and a L ayered Elast ic Sol id ," A SM E

JOURNAL

OF

LUBRI CATION TECHNOLOGY,

A pr. 1974, pp. 250-257.

13 Kannel, J . W., and D ow, T. A ., "E valuatio n of Contact Stresses Between

a R ough Elastic and a L ayered Cylinder," to be presented at the Leeds-L yon

Conference, Sept. 1985.

A P P E N D I X

Development of Shear-Deflection Equations

Fourier Transform Equation. Th e objective of this

analysis is to develop a relationship between surface shear

stresses and tangential deflections in absence of applied

normal stresses at the boundary. The analyses are based on

elasticity theory using the Fourier transform approach given

by Sneddon and Gupta and Walowit . These equat ions are

given in the following form (see Fig. A l) .

d

2

^ I f

0 0

= \ c o

2

Gexp(- f'cox)da>

2ir J -oo

x

2

d

2

V

dy

2

d

2

V

J

d

2

G

dy

2

exp(- /cox)a ' co

dxdy 2ir J -

:

d G

t

w

(co

ex p( - /cox) aco

dy

( A l )

l-v

2

(" fd

3

G (2-v)\

2

dG-\ .

v = \ n (

)

u exp ( - /cox)

2TTE J-ooLcfy

3

\ l-v / dy 1

doi

u =

2TTE

p [d

2

G / v \ , -1. , . du

whe r e

V i s the

A i ry s t ress funct ion which sa t i s f ies

t h e

b i ha r mon i c e qua t i on a n d G i s the Fou r i e r t r a n s f o r m of Sf,

t ha t is

V

4

^

= 0

(A2)

Fig.

A1 Coo rdinate system for shear stress analysis

oo

$ exp(, wx)dx (A3)

Eliminating ^ from the above two_ equations and solving the

resulting differential equation in G, we get a solution of the

form

G= A+By)exp - \u\y) + (C+D y)exp(\o)\y)

(A4)

whe r e

A , B , C ,

a n d

D

a r e

c ons t a n t s

o f

i n t e g r a t i on

to be

e va l ua t e d a tt he bo unda r y .

Boundary Condit ions

The bounda r y c ond i t i ons fo r the t rac t ion analys isa r e :

1) bou nda ry s t ress i s t h eapp lied shear stress

2) stress a n d def lec t ions a r e c on t i nuous a c r o s s t h e layer

in ter face

3) stress goes

t o

zero a t oo;

at t h esur face ,

d

2

*

dxdy

dG

p

= -I 1 /'co exp(iu>x)du

2 i r J - = dy

(A6)

,dG

If is an odd function, Sneddon shows that:

dy

1 f

IT

JO

d G

J

co cos wxaco

dy

Based on Fourier Transform theory ;

dG

dy

o

co s

OJ Xdx

(A7)

(A8)

a t t he s ur fa c e v = 0 r = - r

0

. Let t ing T

0

be defined over the

interval A x and lett ing T

0

= 1/Ax then lim we have

dG

c o =1 (A9)

dy

L et t ing

s = hu, G

=

G/h

2

, r\ - y/h,

the first boun dary

condition becomes

dG

, f n

57- =

1

fo r n= 0

d{

(A10)

( A l l )

and

A lso for the case of no norma l stress on the surface

G = 0

Equat ions (A 4) , (A 10), and (A l l ) com bine to y ield

^ , + C , = 0

- ^ 4

1

5

2

+ 5

1

s - t - C

1

i '

2

+ i ?

1

i ' =l (A12)

where the co nstant in this equation are defined in body 1 of

Fig. 1.

4 0 8

/V ol . 108, JULY 1986

Transactions of the ASME



7/7

The remaining boundary conditions can be met using the

same approach as used for the normal stress conditions [12].

These conditions yield the matrix given in Table 1.

Green's Function for Shear

The tangential deflection on the surface from equ ation ( A l)

can be expressed as:

( l - i >

2

) I"

30

d

2

G cos if

7rE

I;

d

v

2

'-ds

where

d

2

G

~ ~ h f

-2(5, -D

{

s

(A 13)

(A 14)

The Green's function can be written with reference to an

arbitrary displacement

V

x

at f = 1

1 _ 2

u

u,

=

r f H L

2 ( 5

'

-D,)s

co ssi;- co ss

efe-2j31nf]

(A 15)

It can be shown, using the matrix in Table 1 that for larger

values of s, (B, - > , ) - \/s. Equ ation (A 15) can be ex

pressed as two integrals (0 < s < s

0

) an d (s

0

< s), as given in

the text.

If we let

then this equation correspo nds to eq uation (5) in the text.

Because the shear deformation is calculated as a relative

displacement, a reference point must be selected. The

displacements c alculated from equation (A 15) becomes sm all

for large f; therefore a reasonable assumption for the

reference point is 5 contact widths (x = -1 0 6 ) . Th en t h e

relative ta nge ntial deflection (e,-) is:

e

Analysis of Traction Forces in a... Kannel 1986

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