Design of a kinematic coupling for precision applications

ELSEVIER

Design of a kinematic coupling for precision applications C. H. Schouten, P. C. J. N. Rosielle, and P. H. J. Schellekens Department of Mechanical Engineering, Precision Engineering, Eindhoven University of Technology, Eindhoven, The Netherlands To m a c h i n e a complex precision product, several tools are n e e d e d . These tools are placed on a tool turret. A tool must r e t u r n several times to its original position. To attain a very high repeatability between the u p p e r part a n d the base o f the tool turret m o u n t e d on a precision lathe, it is preferable that the parts of the tool turret are statically d e t e r m i n e d in their contacts. This is attained by using a kinematic coupling. To attain the r e q u i r e d stiffness this coupling is provided with a preload o f 1.5 • 103 N. The machining forces are typically less than 1 Newton. A special kinematic coupling, consisting o f grooves a n d balls, was designed, made, and tested. By providing the grooves with self-adjusting surfaces, hysteresis is r e d u c e d to less than one-tenth o f a micrometer. M a x i m u m stiffness is a i m e d at by using c e m e n t e d carbide, a material with a high admissible stress, at the contact points. Experiments show that this kinematic coupling, u n d e r a preload o f 1.5 • 10 3 N, has a static stiffness o f m o r e than 1 • 10 8 N / m in every direction and a repeatability better than one-tenth o f a m i c r o m e t e r . © Elsevier Science Inc., 1997

Keywords: kinematic coupling; repeatability; static stiffness; hysteresis; self-adjusting surfaces; elastic hinges

Introduction The position o f a r/g/d body is fully d e f i n e d if all six degrees of f r e e d o m are constrained; t h e n the body is said to be "statically d e t e r m i n e d . " Constraining a d e g r e e of f r e e d o m m o r e than once requires elasticity (so the body c a n n o t be r/g/d any m o r e ) , which introduces difficulties in the design. It is, therefore, preferable to design statically determined, especially for precision machines, a,2 With such a statically d e t e r m i n e d body, it is possible to replace that body to the original position, with respect to the rest of the machine, with high repeatability. T h e r e are no r e d u n d a n t elements that constrain the same d e g r e e o f freedom. Repeatability of such a m a c h i n e can be m u c h better than the m a n u f a c t u r i n g accuracy. In the literature, such statically d e t e r m i n e d bodies are known as "kinematic couplings "3-5 or Address reprint requests to Ir. C. H. Schouten, Tinelstract 166,

5654 LX, Eindhoven, The Netherlands.

Precision Engineering 20:46-52, 1997 © Elsevier Science Inc., 1997 655 Avenue of the Americas, New York, NY 10010

"Kelvin clamps. ''6 A kinematic coupling consists of two parts. The n u m b e r of contact points between these two parts equals the n u m b e r of degrees of f r e e d o m that must be constrained. O n e of the possibilities is a kinematic coupling consisting of three radial V-grooves in one part and three balls (or segments of balls) on the o t h e r part, as shown in Figure 1. Every ball has two contact points with a V-groove. This design consists o f similar parts, and it shows predictable thermal behavior, with the thermal c e n t e r located in the middle o f the coupling. A high stiffness is obtained if the coupling is provided with a preload. T h e value of the maxim u m static stiffness that will be attained d e p e n d s u p o n such parameters as the admissible stress of the material and the dimensions of the balls and grooves. If these parameters are known, the value o f the static stiffness is highly predictable. 7 Because o f the symmetry of the coupling, a central preload in the vertical direction causes a similar deflection o f every contact. Calculations are often based on the assump0141-6359/97/$17.00 PII S0141-6359(97)00002-0

S c h o u t e n et al.: Kinematic coupling for precision applications

Inthedirection ofthepreload

,.sioo:r 2sin2aC

/~2cosZa Cn

Figure 2 Schematic diagram of a ball in a conventional V-groove

Figure 1 A kinematic V-grooves and three balls

coupling

with

three

tion of ideal Hertzian contacts 3,5 between the bodies. In this context, "ideal" m e a n s no friction between the bodies in contact and no surface roughness. In this frictionless situation, contact stresses do not c h a n g e if there is a relative movem e n t between surfaces. However, in real contacts, every relative m o v e m e n t between surfaces introduces a friction force Fp in the direction opposite to the m o v e m e n t . This friction force is tangential to the surface a n d causes a tangential deflection 8t. Expressions for the tangential displacement have b e e n f o u n d by Mindlin for the case of no-slip a n d by Deresiewicz for partial slip ( r e p o r t e d in J o h n sonS). In the first instance there is no slip at the interfaces. T h e tangential deflection 8 t is a function of the tangential friction force Ff, the radius o f the contact area a, the shear modules G, a n d the Poisson ratio v of the contact material, a T h e tangential stiffness is: F~ ~t -

ct-

4aG 2 -

Cn

-

2. IF, I Sv -

with

G-

2(1 + v)

(1)

2(1 - v)

(2)

2 -- V

Contact between a ball and a conventional V-groove is shown schematically in F i g u r e 2. Variation of the preload causes a variation in (elastic) d e f o r m a t i o n (symbolized by an elastic spring) in the contact. It is also possible that there is a movem e n t between the surfaces (symbolized by friction). W h e n the preload varies so m u c h that there is a relative m o v e m e n t between ball a n d V-groove, PRECISION ENGINEERING

(3)

C

In this equation, F f is the friction force and c the stiffness in the same direction. If the preload on the ball in the V-groove varies, the m a x i m u m friction force that will a p p e a r is Ft. = f " F,, (with coefficient of friction f and reactive force F,, in the contacts n o r m a l to the surfaces of the V-groove). By using this relationship for ~ in Equation (3), m a x i m u m hysteresis in the direction of the preload is given by: sv -

2" [ F,,~I cn,~

2 " 12"

f"

Fn" sine[

2 • cos2cx • cn

(4)

In this equation, Ff, is the friction force in the direction of the pre~'oad and %/, the n o r m a l stiffness in the same direction, Figure 2. With F,, = F p / 2 c o s e ~ , Equation (4) becomes:

E v

With a "local" n o r m a l stiffness c n =- d F n / ~ , , o f E e • a (Hertzian theory) a n d an equivalent elastic Young's m o d u l e s E e = E / ( 1 - v ~) Equation (1) becomes: Ct

it is almost certain that the original position will n o t be regained once this variation no l o n g e r exists. This uncertainty of position is t e r m e d hysteresis s,,1 a n d can be described as:

f" F0 • tano~ Sv-

(5)

COS20~" C n

For o u r application, repeatability needs to be better than one-tenth of a m i c r o m e t e r . Hysteresis is very undesirable w h e n such a high repeatability is required. Equation (b) shows that hysteresis is small if the angle of inclination c~ is small a n d / o r the coefficient of friction f is small a n d / o r the n o r m a l stiffness c,, is large for a certain preload FF Hysteresis is theoretically zero if there is no relative m o v e m e n t between the ball and the surfaces of the V-groove. Minimizing

hysteresis

Repeatability improves w h e n hysteresis decreases, because t h e n the indefiniteness o f position is less after a force or t e m p e r a t u r e variation. According 47

S c h o u t e n et al.: Kinematic coupling for precision applications

I

hin~ saw-cut~

lastic

Z•x Y

Figure 3 Elastic hinge principle to Equation (5), this can be realized by, for example, the choice of a suitable material for the contacts: (1) with a low coefficient of friction f; a n d (2) with a high admissible (Hertzian) stress (for a high feasible stiffness %). A high value of the normal static stiffness leads n o t only to m i n i m u m hysteresis, b u t also to a high total stiffness of the coupling. That is why c e m e n t e d carbide, a material with a very high admissible stress ( ~ 5 • 103 N / m m 2 ) , is used at the contacts. Because this c e m e n t e d carbide is used, instead of steel, the m a x i m u m contact stiffness that will be attained with the same contact geometry and p r e l o a d /~ is approximately twice as high. 7 Unfortunately, the coefficient o f friction of c e m e n t e d carbide (~-- 0.2) is similar to the coefficient of friction of steel. So hysteresis decreases, b u t if the p r e l o a d varies so m u c h that there is a relative m o v e m e n t b e t w e e n ball and surface of the groove, there is still the same friction force. Friction b e t w e e n the balls a n d grooves is theoretically zero when the surfaces of the V-grooves follow the m o v e m e n t of the balls, when the p r e l o a d increases or decreases. T h e n there is no relative m o v e m e n t b e t w e e n balls and surfaces of the grooves. This can be achieved by providing the V-grooves with selfadjusting surfaces, by means of elastic hinges in the material b e n e a t h the surfaces o f the V-grooves. Such elastic hinges are created by drilling two holes next to each o t h e r in the material and making saw-cuts from the surface to these holes, as shown in Figure 3. In Figure 4 the V-groove-ball situation is depicted, showing the elastic hinges in the material b e n e a t h the surfaces o f the V-groove. The situation is simplified by assuming that the left elastic hinge has a very low rotational stiffness k0 (0 means a rotation a r o u n d the z-axis) and the right half o f the V-groove has stiffnesses c,, a n d ct in the ball-groove-contact and stiffnesses of the elastic hinge; cx normal to the surface and cy tangential to the surface. Because o f the very low rotational 48

stiffness o f the left elastic hinge, the tangential deflection S t in is also very low; whereas, the tangential stiffness ct (= F J ~ t ) in the left contact is very high. If a is approximately 45 °, the m o v e m e n t u 0 o f the left surface of the V-groove by m e a n s o f p u r e rotation of the elastic hinge almost equals the normal deflection ~,,,x in the contact of the ball with the right half o f the V-groove. O f course ct is n o t infinite, b u t because o f the m o v e m e n t o f the surface with the m o v e m e n t o f the ball, the tangential force F¢ is very small a n d so is the d e f e c t i o n S t. For a certain ball diameter in contact with flat surfaces o f the V-groove, the m a x i m u m contact stiffness, normal to a surface o f the V-groove Cn.nla,, is calculated with the help of the Hertzian theory. So the stress in the contacts equals the admissible stress o f c e m e n t e d carbide. 7 Stiffness of the elastic hinge normal to the surface cx is chosen to be ten times as large as C,,,ma,,. T h e total normal stiffness c,,,x (this is a series c o n n e c t i o n of stiffness c,, a n d Cx) is only 9% less than the normal stiffness witho u t the elastic hinge. With the help of a theory r e p o r t e d by J o h n s o n , 8 a relationship b e t w e e n the tangential a n d the normal stiffness in the contact is derived (Equation (2)). W h e n the m a x i m u m normal stiffness Cn,max is given, it is now possible to calculate the m a x i m u m tangential stiffness c¢.... ,, in the contact. T h e stiffnesses o f the elastic hinge; cx (normal to the surface), cy (tangential to the surface) a n d k0 (rotational stiffness) are calculated with equations given in a university textbook. 1 W h e n the distance b e t w e e n hinge and contact is a, the tangential stiffness in the contact, caused bff the rotational stiffness of the hinge, is co -~ k o / a 2. T h e total tangential stiffness in contact Ct,y,O is a series c o n n e c t i o n o f the stiffnesses ct, cy, and c0. The total normal stiffness almost equals the normal stiffness without the elastic hinge. However, the rotational stiffness has to be low to achieve the r e q u i r e d mobility of the surfaces, so the tangential stiffness (with the elastic hinge) has to be m u c h less than the tangential stiffness without elastic ua= 5n,X"sin 2a

5n~ /

cy

Figure 4 Ball in a V-groove with elastic hinges JANUARY

1997 V O L 20 N O 1

Schouten et al.: Kinematic coupling for precision applications hinge. So, unfortunately, a disadvantage o f a coupling with elastic hinges is that the total stiffness is lower than the total stiffness of a coupling with conventional V-grooves. T h e total axial stiffness Ca,totaI of the coupling with elastic self-adjusting V-grooves can be described with7: Ca,tota~ = 6 " COS2CX " Cn,x + 6 • s i n 2 a • ct, v,o

(6)

The total radial stiffness C,;totaI of this coupling can be described withY: Craota, = 3 " sin2eL(Cn,x -- Ct, v,,) + 3 " (ct, v,e + ct)

(7)

T h e total axial a n d radial stiffness of a coupling with conventianal V-grooves can be described with the same Equations (6) and (7), w h e n c,, and ct are substituted for c,,,x respectively ct,y,o. Because ct < % (with V ~ 0,25 equation [2] becomes: ct ~ 0.86- %) a n d ct,y e << c,~,, (q,ye "~ 0.01 • c,,,x), the axial stiffness cfecreasesl anal' the radial stiffness increases as the angle of inclination 0t increases. So, the best choice for the angle of inclination is if C~;tota I = Ca,tota I. For the kinematic coupling with elastic self-adjusting V-grooves, this is 37 °7. For the conventional coupling it is 550, 7 but this value is less critical, because the tangential stiffness c¢ has the same o r d e r of m a g n i t u d e as the n o r m a l stiffness % Also i m p o r t a n t is the stiffness of the total coupling u n d e r eccentric load (in tangential direction). T h e equation for the so-called total tangential stiffness ct,tot~1 of the coupling with elastic self-adjusting V-grooves can be described withY: Cr, tota I " Ct, tota I

3 • (ca, x" s i n 2 ~ + Ct, y,O" CO S2OL)

cr,tota~ + 3 • (Cn,x" S i n 2or + Ct, v,e" COS 2or)

(8) T h e angle o f inclination oL has to be of such a value that the coupling does not self-lock d u r i n g positioning. Positioning of the coupling with elastic self-adjusting surfaces does n o t give that problem, because of the mobility of the V-groove surfaces. In case of a conventional coupling, the c h a n g e that self-locking does o c c u r is the smallest tbr an angle of inclination o f 43 °7. This angle of inclination is n o t critical. For an angle of 2 9 - 5 6 ° , the admissible coefficient o f friction is m o r e than 0.2 (the coefficient of friction of c e m e n t e d carbide).

Design of a kinematic coupling and test results A special kinematic coupling was designed. O n e part o f this coupling has six V-grooves: three conventional a n d three self-adjusting, all with an a n -

PRECISION ENGINEERING

gle of inclination of 40 °. O n tile c o u n t e r p a r t o f the coupling, consisting of a disc, three (15-mm diameter) ball segments are m o u n t e d equally divided on the same radius. T h e coupling is m a d e of steel; c e m e n t e d carbide is used only at the points of contact. T h e material properties of c e m e n t e d carbide are: an elastic Young's m o d u l e s E o f 6 • 105 N / m m 2, an admissible stress of 5" 103 N / m m 2, Poisson ratio v of 0.25, and a coefficient of friction f o f 0.2. C e m e n t e d carbide tiles, with a thickness of 2 m m , are glued on the surfaces o f the V-grooves, a n d three c e m e n t e d carbide ball segments are glued on the o t h e r part o f the coupling. The glue used is a t h e r m o s e t with an elastic Young's modules of 2 • 109 N / m 2. Because the glue layer is very thin (only a few micrometers) a n d the contact area fairly large (1.8 • 10 - 4 m 2 ) , this layer is very stiff. This design enables comparison between a conventional coupling a n d a coupling with sell= adjusting surfaces (by means of elastic hinges). For this purpose, two holes are drilled next to each o t h e r in the material b e n e a t h the two surfaces of three V-grooves. T h e middle o f the d a m between the holes coincides with the n o r m a l to the surface of the V-groove t h r o u g h the contact (of the ball with the V-groove). Cuts make the surfaces relatively flexible in the tangential direction with respect to the rest of the coupling, f i g u r e 5 is a p h o t o g r a p h of the kinematic coupling. T h e coupling was tested with the help of a testing system, as shown in b~gure 6. By moving a mass M, the kinematic coupling is provided with a varying preload in vertical direction, by way of a beam, two rods, a second beam, a n d finally a ball on the middle of the coupling. T h e relative movem e n t of the disc with the three ball segments with regard to the part with the V-grooves, attributable to the compression in the vertical direction (by elastic deflection o f the contact surfaces), is measured with a displacement t r a n s d u c e r fixed to the u p p e r disc o f the coupling. The preload varies progressively a n d is m e a s u r e d with a d y n a m o m e ter, fixed to the b o t t o m of the lowest beam. T h e d y n a m o m e t e r measures the pulling force on the beam. T h e r e f o r e , the m e a s u r e m e n t s have to be c o r r e c t e d with the static mass o f the disc, the beams, the rods and the ball (a total weight of 3.5 kg).

Elastic deflection and hysteresis T h e coupling should be provided with a load to attain the r e q u i r e d stiffness. This load is called a preload a n d is m u c h larger than the external loads in use, which are typically one Newton. Preload versus displacement is recorded. T h e vertical dis-

49

Schouten et al.: Kinematic coupling for precision applications Total axial stiffness of the coupling

The tangent (for a specific preload) on the prel o a d - c o m p r e s s i o n graphs of Figure 7 gives the axial stiffness of the coupling. If the preload is 1.5 • 10 3 N, the axial stiffness of the conventional coupling is about 6.1 • 1 0 2 N/lxm and for the coupling with self-adjusting surfaces about 3.6" 10 9 N/lxm.

b

~05"0.

Figure 5 (a) The kinematic coupling parts; (b) detail drawing: an elastic hinge

placement is the relative m o v e m e n t between the two halves of the coupling, which is the result of the elastic deflection of the contact surfaces, is plotted on the x-axis. The preload is plotted on the y-axis. In Figure 7, the p r e l o a d - c o m p r e s s i o n graphs for both couplings are shown next to each other. Hysteresis is the difference between the compression w h e n the preload increases and the compression w h e n the preload decreases. As explained earlier, this hysteresis is the result of friction between balls and V-grooves. W h e n the preload decreases, the friction force is in the opposite direction to the friction force with an increasing preload. T h e r e is no relative movem e n t of the bodies in contact until the friction has c h a n g e d direction. Figure 7 shows that hysteresis decreases by a factor of 10 if the surfaces of the V-grooves are provided with these elastic hinges.

50

displacement transducer I v

//////////

~.,

v, I

dynamometer f

/~

L

/

"f

Figure 6 (a) The testing system; (b) diagram of the testing system

JANUARY 1997 VOL 20 NO 1

S c h o u t e n et al.: Kinematic coupling for precision applications

1500

tained if the force in the radial direction is divided by the radial deviation. T h e radial stiffness o f the conventional coupling is a b o u t 6.0" 1 0 2 N / t x m a n d a b o u t 3.7 • 1 0 2 N / t x m for the coupling with self-adjusting surfaces.

7 /

Fp rN]l 1 Conventional V-grooves 2 Self-adjusting V-grooves

//

1000

s0o

/'

Total tangential stiffness of the coupling In the same way as for the radial stiffness, the tangential stiffness of the coupling (stiffness u n d e r eccentric load) is m e a s u r e d with the set up. Now the additional force is in the tangential direction and the deviation is m e a s u r e d in the direction of this force. In Table 1, the results of the m e a s u r e m e n t s a n d the calculated values are shown. All measurements are taken with a p r e l o a d Fp of 1.5 • 103 N, and calculations are m a d e for the same value of the preload. The u n c e r t a i n ~ of the e x p e r i m e n t s was estimated to be +__ 5 • 10 N/Ixm. Calculated values a n d m e a s u r e d values of the axial a n d radial stiffness of the coupling agree well (Equations [6] a n d [7]). These values are highly predictable. It is m o r e difficult to make a g o o d estimate o f the tangential stiffness of the coupling (Equation [8] ). For the conventional coupling, the m a x i m u m value of hysteresis is calculated with Equation (5) at a total p r e l o a d of 7.5 • 1 0 2 N (after a variation from 0 to 1.5 • 103 N). The e x p e r i m e n t a l value of hysteresis of the conventional coupling is less than this calculated value, b u t n o t less than one-tenth o f a micrometer. By providing the grooves with selfadjusting surfaces, hysteresis is r e d u c e d considerably. Because the m a x i m u m value of the friction force F~......~x (= f " F,, ~ 66 N) is m u c h larger than the real tangential force F t ( ~ 3.3 N) hysteresis caused by m a c r o slip is theoretically zero. Results o f the e x p e r i m e n t s show that there still is some micro slip in the contacts.

/ 1

2

3

4

8p [IJm]

Figure 7 Measurements of preload Fpversus elastic deflection Gp; results: hysteresis and axial stiffness Total radial stiffness of the coupling T h e radial stiffness o f the coupling is m e a s u r e d with the same setup. Again, the kinematic coupling is provided with a central p r e l o a d o f 1.5 • 103 N in the vertical direction, as shown in Figure 6. Using a spring-type force gauge, an additional force in the radial direction is i n t r o d u c e d to the b o t t o m of the u p p e r disc (at the height o f the contacts). T h e radial deviation of the disc with respect to the base, which is the result o f the radial force, is m e a s u r e d with a d i s p l a c e m e n t transducer. An a p p r o x i m a t i o n of the radial stiffness is ob-

Table 1 Comparison of theory with experiments for a preload Fvof 1.5.103 N

Axial stiffness, %total, N/t xm Radial stiffness, c,~t,,tal, N/b~m Tangential stiffness Ct,total, N / ~ m Hysteresis, btm

PRECISION ENGINEERING

Conventional V-grooves

Self-adjusting V-grooves

Calculations

Experiments

Calculations

Experiments

6.5

6.1

• 10 2

3.7 • 102

3.6 • 102

6.1 • 109

6.0

• 10 2

4.3 " 10 2

3.7 • 109

3.1 • 102

1.4" 10`2

1.6" 102

1.1 • 109

max 0.79

0.42

None

0.03

" 10 2

51

Schouten et al.: gdnematic coupling for precision applications

Conclusions Hysteresis is very undesirable w h e n repeatability between the u p p e r part a n d the base of a tool turret m o u n t e d on a precision lathe (to r e t u r n a tool several times to the original position) must be better than one-tenth of a m i c r o m e t e r . After an external force or t e m p e r a t u r e variation, hysteresis can occur. By providing the grooves of a kinematic coupling with self-adjusting surfaces, hysteresis is r e d u c e d by 95% and is less than one-tenth of a m i c r o m e t e r for a preload of 1.5 • 103 N. This preload is necessary to attain the r e q u i r e d stiffness in all directions. If it is possible to r e p r o d u c e the preload with high accuracy, repeatability only depends u p o n the stability of t e m p e r a t u r e and expansion of the design. The static stiffness of the coupling is highly predictable, with the equations given in this article, for the conventional coupling as well as for the coupling with self-adjusting surfaces. If the preload is as large as possible, which d e p e n d s u p o n the admissible load of the material, the stiffness o f both kinematic couplings are sufficient for most purposes,

T u

x Y z

Greek 0~ 5 0 v ~p

a e f F max n P r t

total x

y

The e x p e r i m e n t s were carried out in the m e c h a n ical d e p a r t m e n t of the Philips Research Laboratories Eindhoven. We thank all contributors, especially F. Jaartsveld a n d J. J. Baalbergen, for their advice, help, and skill, which were essential to this project.

z 0 ~P

a a c

D E

f

F G h k M R Sv t

52

Radius o f a circular contact area, m m Arm, distance contact-hinge, m m Stiffness, N / p . m Diameter, m m Elastic Young's modules, N / m m 2 Coefficient o f friction, [ - ] Force, N Shear modules, N / m m ~ Dam, m, m m Rotational stiffness, N m / r a d Moment, Nm Radii of curvature, m m Virtual play (hysteresis), p.m Thickness, m m

Angle of inclination, [°], rad Compression (elastic deflection), /xm Spin parameter, rad Poisson ratio, [ - ] Spin parameter, rad Spin parameter, tad

Subscripts

Acknowledgments

Notation

T e m p e r a t u r e , °C, K Movement, m, m m Axis o f coordinates, m, m m Axis of coordinates, m, m m Axis of coordinates, m, m m

Axial Equivalent Friction o f the force Maximum Normal PreRadial Tangential Total in direction o f x in direction of y in direction of z Rotation a r o u n d z Rotation a r o u n d x Rotation a r o u n d y

References 1 Design Principles. Eindhoven, The Netherlands: Eindhoven University of Technology (no. 4007), 1996 2 Slocum A. H. Precision Machine Design. Englewood Cliffs, NJ: Prentice-Hall, 1992 3 Slocum A. H. Kinematic couplings for precision fixturing-Part 1--Formulation of design parameters. Prec Eng, 1988, 10, 85-91 4 Slocum A. H., Donmez A. Kinematic couplings for precision fixturing--Part 2--Experimental determination of repeatability and stiffness. Prec Eng, 1988, 10, 115-121 5 Slocum A. H. Design of three-groove kinematic couplings. Prec Eng, 1992, 14, 67-73 6 Sherrington I., Smith E. H. Design of a Kelvin clamp for use in relocation analysis of surface topography. Prec Eng, 1993, 15, 77-85 7 Schouten C. H. "Het ontwerp van een precisiegereedschapwisselaar .... Design of a precision tool-changer", Master's thesis (in Dutch), Eindhoven University of Technology, Eindhoven, The Netherlands, 1994 8 Johnson K. L. Contact Mechanics. Cambridge, UK: Cambridge University Press, 1992

JANUARY 1997 VOL 20 NO 1