The Delft EHL diagram from a historic perspective: righting a 50-year-old wrong

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The Delft EHL diagram from a historic perspective

Citation for published version (APA):

van Leeuwen, H. J. (2018). The Delft EHL diagram from a historic perspective: righting a 50-year-old wrong. In

E. Kuhn (Ed.), 13. Arnold Tross Kolloquium, 15 May 2017, Hamburg, Germany (pp. 120-157). Shaker-Verlag.

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Published: 01/01/2018

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The Delft EHL Diagram

from a historic perspective

Righting a 50-year-old wrong

13th_{Arnold Tross Colloquium}

Hamburg, May 19th_{, 2017}

Harry van Leeuwen

Presentation Outline

1 Introduction and aim 1. Introduction and aim

2. A primer in nondimensional numbers

3. Nondimensional grouping in stationary EHL line contact 4 Consequences for EHL groups

4. Consequences for EHL groups 5. Results

6. Discussion

7 C l i

7. Conclusions

/ Department of Mechanical Engineering

Combustion Technology Group PAGE 1

1. Introduction and aim - 1

Motivation :

Since many years nondimensional groups have been used in stationary EHL line contact theory. The so-called Moes groups have become standard in most of the literature, after its first appearance in

Proceedings of the Instn. Mech. Engrs., 1965-66, Vol. 180, Pt. 3B. Three

nondimensional groups were established: H, L and M.g p ,

These groups were to replace the four groups defined by Dowson and Higginson in their publications and book Elastohydrodynamic

Most people nowadays refer to these 3 groups as the “Moes groups”,

gg p y y

Lubrication (years 1959-1966), H, U, W and G.

and even a lot more scientists prefer to use the 4 Dowson and Higginson groups.

Why is this so?

The following questions arise: • Why is less better than more?

y

• Why is less better than more?

• Why do most authors persevere in using 4 groups? • How did research results in tribology proliferate?

• Was Moes the first to define and use these nondimensional EHL groups?

Therefore subsequent topics will be treated • a strategy to find nondimensional numbers • applied to EHL stationary line contactsapplied to EHL stationary line contacts

• mapping data from one group set into another set • the original work as a mark of time

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Inducement

Greenwood (1969) writes: ““…

Clearly there are many equivalent sets of three groups and the choice is largely personal: on the other hand, the use of four choice is largely personal: on the other hand, the use of four groups may be dangerous as it permits equations to be written which are incompatible with the basic equations: any valid solution can be written in terms of Blok's three groups.g p …”

6. Discussion

7 C l i

7. Conclusions

2. A primer in nondimensional numbers

Why nondimensional numbers (nondimensional groups)?

 are independent of any system of units, e.g. m-k-s or f-p-s system, provided that it is a consistent system. This means that all quantities in the numbers can be expressed in the dimensions fundamental to the system, and p y , that nondimensional numbers are independent of the choice of the system,  allow a minimization in the number of parameter groups that describe the

problem, resulting in minimizing the number of numeric or experimental problem, resulting in minimizing the number of numeric or experimental simulations. More to follow.

 well-scaled nondimensional groups define a ratio or scale between two physical effects

physical effects

 well-scaled physical problems also result in less trouble finding a numerical solution.

 allow an easy and quick check on the punctuality of the groups by a dimension check

PAGE 6

2. A primer in nondimensional numbers - 2

How can nondimensional groups be found?

1) Define the physical-mathematical model by a set containing all equations, including initial and boundary conditions

Method developed by Blok (1958) and perfected by Moes (1992)

including initial and boundary conditions

2) Render all equations nondimensional by splitting up each variable in a capped part “^” containing the dimension, and a nondimensional barred part “¯”

3) Substitute an identifier for nondimensional parameter groups, e.g., Nifor a

group numbered ‘i’

4)) Determine the number of caps “^” to be arbitrarily chosenp y

5) Choose as many groups N_i= 1 if there are variables that are free to choose.

6) This yields a compulsory choice for all overcapped quantities, and for any remaining parameter groups

remaining parameter groups

7) The remaining groups are the parameters for the problem that determine the solution.

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In our case of stationary EHL line contacts:

dh U = ) dp (h d 6 3 _ _ Reynolds’ equation: 0      p x

with boundary conditions:

where dx U = ) dx (h dx 6 sum 0 0       dx dp p x x p x cav



U U



u U where sum 1 2  2

 

p p







( ) 0exp

film geometry including deformation

fluid viscosity by Barus’ relation

 

ds s s x s p E R x h = x h 2 2 0 ln 2 2 ) (          _ 

 

p p



( ) 0 p

film geometry including deformation

load equilibrium l ti it f d l d d E curvature of radius reduced = R where s E R r r r 2    



     x x dx x p l F ) (

elasticity of modulus reduced = Er

Introduction of nondimensional variables:

ˆ

h

p

ˆ



s

x

ˆ



where barred “¯”variables are nondimensional, and capped “^” ones dimensional.

When these substitutions are performed:

   __      pp som x d h d h p x U x d p d e h x d d ˆ ˆ ˆ 6 2 0 ˆ 3          p x x d h p x d x d 0     cav x d p d p x x x 0 ˆ

 

           p s xx ss ds E h s p x R h x h h = h where ˆ ˆ ˆ ln ˆ ˆ ˆ 2 ˆ 2 ˆ ˆ 2 2 2 0

 

     r r and s s E h R h h 2       x x x d x p l x p F ) ( ˆ ˆ

 __      Np x d h d N x d p d e h x d d 6 2 3 1 x U N p N sumˆ ˆ 0 2 1          p d p N p x 0 0 x x N h p N cav ˆ ˆ ˆ 3 2 2      where x d p p N x ₃ 0 x N h h N ˆ ˆ 2 5 0 4  

 

            ds s s x N s p N x N N = h ln 2 7 6 2 5 4 E h s p N R h N r r ˆ ˆ ˆ 2 ˆ 2 6 5      x p xdx N and ) ( 8 _N F s x N r ˆ ˆ 7       x p( ) 8 l x p N ˆ ˆ 8 where 4 out of 8 groups may be chosen fully arbitrarily, because the 4 capped variables may be chosen freely

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6. Discussion

7 C l i

7. Conclusions

3. Nondimensional grouping in EHL line contact

So choose e.g.: ˆ ˆ 1 ˆ 2 ˆ 1 2 * 5   R h x N r _l_R E F p r r  4 ˆ* ˆ ˆ 1 2 ˆ 1 * 7 * 6       s x N E h s p N r  hence: l E F h r  2 ˆ*_ 1 ˆ ˆ 1 1 * 8 7       l x p F N s x N l E R F s x r r  4 ˆ ˆ* *

Up till now no knowledge of contact mechanics has been used. If it is now realized that for Hertzian line contacts

l E R F b and R l E F r r Hz r r Hz _ _  8 2  

3. Nondimensional grouping in EHL line contact - 2

So apparently a physically appealing set is found if

ˆ 2 ˆ2 * * 5      R h x N r H r E F pˆ**  2 ˆ ˆ ˆ 1 * * * * 6     E h s p N r yielding: Hz r l E F h R l p      ˆ 2 * * 2 ˆ ˆ 2 ˆ ˆ 1 * * 8 * * 7   _ _     l x p F N s x N Hz r r r b l E R F s x l E     8 ˆ ˆ** ** 2 2 pxl

PAGE 14

The remaining groups now are

E F N r 2 _Hz 1 2 1 _         E U R l E U N R l r sum r r sum Hz r 4 0 2 2 0 2 1 1 4 2                    b x R F l E x N R E F R E H cav r cav Hz r r r r 3 2 2 8              F l E h N b R F r Hz r 0 4 8 

where N3defines the position of the cavitation boundary, and N4the film

height at x =0, so N4determines the film profile.

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The equations now are

) 1 ( 6 ₂ 3 1 _      N p x d h d N x d p d e h x d

d Note that there are essentially

2 equations:

•Reynolds’ equation(1) and

) 2 ( 0 0          p d p N x p x

Reynolds equation (1), and •the boundary conditions (2)

0 2 3            where x d p N x

 

ln 2 2 4           

_

   and s d s s x s p x N = h 2 ) ( 



     x x x d x p and

This means that



1 2



3 3  cav , Hz cav l E h N N N x b x N

N₁and N₂define the

location of the onset of cavitation N3(xcav) and the

non-

1 2



4 0 0 4 r   , h N N N h F l E h N 3( cav) dimensional filmthickness N₄ (h₀at x=0) 2 1 2 1 _       r Hz E F N where   

Hence they define the entire solution to the equations (1)

d (2) hi h i l d th

Hence they define the entire solution to the equations (1)

d (2) hi h i l d th 4 0 2 2 0 2 1 1 4 2                   r sum r r sum Hz r E R E U F R l E R E U N R l   

 and (2), which includes the

nondimensional minimum film thickness hh m i nm i n and the and (2), which includes the

nondimensional minimum film thickness hh m i nm i n and the

2 _ _   r r Hz r rR F E R E   _{nondimensoinal maximum} film pressure .

p

m a xm a x nondimensoinal maximum film pressure .

p

m a xm a x

This implies





min min r F l E h h _{NOTE that}

N is the only group which



1 2



min min , where N N h

h N1is the only group which

contains the pressure viscosity coefficient



, and

N is the onl gro p hich

2 1 2 1 2 _        Hz r r R l E F N   

N₂is the only group which contains the hydrodynamic parameters



₀U_sum, 4 0 2 2 0 2 1 4 _                Hz r r r sum r r r r sum r E R E U F R l E R E U N   

 and thatdoes not contain any

of these: m i n h



₁ ₂



min min

h

N

,N

h



So the effect of



and 0Usumon film thickness can be made in one single twodimensional diagram twodimensional diagram





min min r F l E h h



1 2



min min , where N N h h 2 1 2 1 2 _        Hz r r R l E F N    4 0 2 2 0 2 1 4 _                Hz r r r sum r r r r sum r E R E U F R l E R E U N    

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3. Nondimensional grouping in stationary EHL line contact 4. Consequences for EHL groups

5. Results 6 Discussion 6. Discussion 7. Conclusions

4. Consequences for EHL groups

Creating custom sets of nondimensional groups

Because the set which has been found is a consistent set, any new set can be created by permutations of the set arrived at. Hence it is possible to create new consistent sets having different properties, e.g. where the influence of the elasticity effect of the solid and the pressure viscosity effect of the fluid are explicitly shown.

This conversion from one set to a new set can be performed as follows. We have found min i  l E h h r We can transform min 1 h E l _N 11 _N12 const h _ __ r __ p _ p   2 1 2 2 1 min min min ,       E F N N h h F h  2 1 1 2 1 min 2 1 32 31 22 21 12 11 N N const N N N F const h p p p p p p   _ _ _         2 0 2 2 1 4 2                  F R l E R E U N R l E F N r r som Hz r r     



1 2



min min 2 1 2 , 3 31 32 N N h h N N const N p p     _  

where p_ijare exponents free to choose



 F

R

Er r where pijare exponents, free to choose

4. Consequences for EHL groups - 2

If we now choose, e.g., i const (i) pi1 pi2 1 8 0 -1 2 22 0 - ½  i const (i) pi1 pi2 1 2 0 - ½ 2 2 0 - ½ 3 4 1 - ½

… then new sets for EHL line contacts are found

3  1 ¼ 2 1 2 0 min 2 2 F l R U hF h r sum         2 1 0 min F R E U R E R h h sum r r r              2 1 2 3 2 2 2 0 1 2 2 g F N g l R E U F N v E r r sum  _                      4 1 0 2 0 1 L R E U E N M l R E F U R E N sum r r r sum r r                            Johnson (1970) Moes (1966)



1 2



min min 3 2 2 ,N N h h g l R U N v r sum         

_

_

2 1 min min h N,N h R Er r     _    

The underlying data are exactly the same the representation depends on what

PAGE 22

The underlying data are exactly the same, the representation depends on what this representation is aiming for.

The relationships between common groups in EHL line contacts are given in a conversion table:

conversion table:

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The original film thickness diagram by Dowson and Whittaker (1964) from the Book by Dowson and Higginson (1977) immediately shows the limitations: Book by Dowson and Higginson (1977) immediately shows the limitations:



UW G



H H





r R h H G W U H H  , , r r F R E u U           0 r r r E G l R E F W         

The results of the

t f ti

Source: Dowson and Higginson (1977) p. 101

r

transformations will be shown in the next section

5. Results

6 Discussion 6. Discussion 7. Conclusions

5. Results

For the representation in groups favoured by groups favoured by Johnson (1970)





min min h g ,g h _E _v 2 1 2 0 min 2 2    F l R U hF h r sum  2 1 3 2 2 2 0 2 2 2            F l R E U F g r r sum E  2 3 2 3 2 2          l R U F g r sum v  

Source: van Leeuwen and Schouten (1995), p. 18

5. Results - 2

For the representation in groups favoured Moes (1966)





   R E h M L H H 2 1 ,          U U R E R h H sum r r r 4 1 2 0 

 

                F R E M R E U E L r r r r sum r 4 0            l R E U M r r sum 0  Source: Courtesy Delft University, 1969

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5. Results - 3

For the representation in groups showing surface speed and load effects:



,



~ ~  h g g h h u F





2 1 0 3 4 2 ~      u E g E R h h r r r    2 1 2 2 0             F E R g r r r u         l R F E g r r F  Source: van Leeuwen et. al. (1987)

5. Results

6 Discussion

6. Discussion

7. Conclusions

5. Discussion

A much older publication exists, where all of this looks very familiar: A closer look reveals:

K,N H H    2 1 0 H const U R E R h const N h const H sum r r r              2 3 2 1 0 1 3 l R E F U R E E K r r sum r r r                 2 2 0 2 1 2 3 2 M l R E F U R E N g M L r r sum r r v                   Source: Peppler (1957) 0UsumErRrl 

PAGE 30

(1957)

5. Discussion - 2

Peppler (1957) reproduced this figure with permission of the author, Harmen Blok from Delft University, who had collected most of the numerical results of that time into one single and experimental survey diagram.

This makes clear that:

Blok freely shared his research results with his colleagues, and they were able to publish them sooner than he did himself (with proper credits)

Those times were a lot different from nowadays where publication pressureThose times were a lot different from nowadays, where publication pressure and competition between universities is common.

Already before 1957 Blok had developed a consistent set of 3 nondimensional numbers to describe EHL film thickness behaviour nondimensional numbers to describe EHL film thickness behaviour

These nondimensional groups developed by Blok were essentially the same as the groups used later by his students Koets (see Dowson and Higginson, (1966)) and Moes (1966)

( )) ( )

 Blok preferred to designate the EHL film thickness chart with these nondimensional groups as the Delft Diagram.

After 1957 Blok replaced the letter ‘N’ by the letter ‘M’, to honour his Swiss

PAGE 31

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5. Discussion - 3

So

the letter ‘M’ is not related to the name of “Moes”, but to “Meldahl”

there is no longer a reason to refer to these groups as the nondimensional Moes numbers!

the name of the nondimensional chart preferably is the Delft Diagram and

if one wants to associate a name with these nondimensional groups, the best choice would be Blok goupsg p

5. Results 6 Discussion 6. Discussion

7. Conclusions

6. Conclusions

Conclusions

 It was found that film thickness in stationary line contact EHL can be described by a set of merely 3 nondimensional groups

 The choice of any of all possible sets depends on which effects should The choice of any of all possible sets depends on which effects should be emphasized

 The EHL film thickness survey chart should preferably be called the Delft Diagram, and

Diagram, and

 The nondimensional groups should be called after Harmen Blok

Acknowledgements:

I would like to acknowledge Hans Moes for being an extraordinary good teacher to me, and Harmen Blok, the professor from Delft, a giant in tribology, to whom all Dutch tribologists owe so much.

g gy, g

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References:

Blok, H., 1958, “Application of dimensions analysis in the theory of heat transfer”, report of a lecture for the Heat Transport Section of the Royal Institute of Engineers, Utrecht

Dowson, D., and Higginson, G.R., 1966, Elasto-hydrodynamic lubrication, Pergamon, Oxford, 1st

Dowson, D., and Higginson, G.R., 1966, Elasto hydrodynamic lubrication, Pergamon, Oxford, 1 edition, 235 pp., and 1977, 2nd_{and SI edition, 235 pp.}

Greenwood J. A., 1969, “Presentation of elastohydrodynamic film thickness results” Instn Mech.

Engng Sci., 1969, Vol. 11, No. 2, pp. 128–132

Johnson, K.L., 1970, "Regimes of elastohydrodynamic lubrication", Journal of Mech. Engineering

Science, Vol. 12, No. 1, pp. 9 -16.

Moes, H., 1992, “Optimum similarity analysis with applications to elastohydrodynamic lubrication”,

Wear, Vol. 159, No. 1, pp. 57-66 Wear, Vol. 159, No. 1, pp. 57 66

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