Engineering Tribology 2011 Part 8 pptx

Thegeometry of parallel cylinders in contact is shown in Figure 7.10 and the formulae for themain contact parameters are summarized in Table 7.2.. TABLE 7.4 Approximate formulae for cont

It can be noted that for the spheres:

R ax = R ay = RA and R bx = Rby = R B

where:

R A and R B are the radii of the spheres ‘A’ and ‘B’ respectively.

Substituting into equation (7.2) gives:

deflection ‘δ’ are presented in the slightly altered form:

Find the contact parameters for two steel balls The normal force is W = 5 [N], the radii

of the balls are RA = 10 × 10-3 [m] and RB = 15 × 10-3 [m] The Young's modulus for bothballs is E = 2.1 × 1011 [Pa] and the Poisson's ratio of steel is υ = 0.3

Since Rax = Ray = RA = 10 × 10-3 [m] and Rbx = Rby = RB = 15 × 10-3 [m] the reduced radii of

curvature in the ‘x’ and ‘y’ directions are:

1 − 0.3 2 2.1× 10 11 ⇒ E' = 2.308× 10 11[Pa]

· Contact Between a Sphere and a Plane Surface

The contact area between a sphere and a plane surface, as shown in Figure 7.9, is also circular.The contact parameters for this configuration can be calculated according to the formulaesummarized in Table 7.1

The radii of curvature of a plane surface are infinite and symmetry of the sphere applies so

that R bx = Rby = ∞ and Rax = Ray = R A The reduced radius of curvature according to (7.2) istherefore given by:

Find the contact parameters for a steel ball on a flat steel plate The normal force is

W = 5 [N], the radius of the ball is RA = 10 × 10-3 [m], the Young's modulus for ball andplate is E = 2.1 × 1011 [Pa] and the Poisson's ratio is υ = 0.3

Since the radii of the ball and the plate are Rax = Ray = 10 × 10-3 [m] and Rbx = Rby = ∞ [m]

respectively, the reduced radii of curvature in ‘x’ and ‘y’ directions are:

· Contact Area Dimensions

· Contact Between Two Parallel Cylinders

The contact area between two parallel cylinders is circumscribed by a narrow rectangle Thegeometry of parallel cylinders in contact is shown in Figure 7.10 and the formulae for themain contact parameters are summarized in Table 7.2

TABLE 7.2 Formulae for contact parameters between two parallel cylinders

b=( )4WR' πlE' 1 /2

rectangle

p max= πbl WElliptical pressure distribution

Maximum contact pressure

Maximum deflection

l is the half length of the contact rectangle [m];

R ' is the reduced radius of curvature for the two parallel cylinders in contact [m]

For the cylinders: R ax = R A , R ay = ∞, Rbx = R B , R by = ∞ where ‘RA ’ and ‘R B’ are the

radii of the cylinders ‘A’ and ‘B’ respectively.

Substituting into equation (7.2) yields:

· Reduced Radius of Curvature

Since the radii of the cylinders are Rax = RA = 10 × 10-3 [m], Ray= ∞ and Rbx = RB = 15 × 10-3

[m], Rby= ∞ respectively, the reduced radii of curvature in the ‘x’ and ‘y’ directions are:

τmax = 0.304p max = 0.304× 55.4 = 16.8 [MPa]

z = 0.786b = 0.786× (5.75× 10 −6 ) = 4.5× 10 −6[m]

· Contact Between Two Crossed Cylinders With Equal Diameters

The contact area between two cylinders with equal diameters crossed at 90° is bounded by acircle This configuration is frequently used in wear experiments since the contact parameterscan easily be determined The contacting cylinders are shown in Figure 7.11 and the contactparameters can be calculated according to the formulae summarized in Table 7.1

Circular contact area

a

R B W

which is the same as for a sphere on a plane surface

If the cylinders are crossed at an angle other than 0° or 90°, i.e their axes are neither parallelnor perpendicular, then the contact area is enclosed by an ellipse Examples of the analysis ofsuch cylindrical contacts can be found in the specialized literature [14] The formulae forevaluation of parameters of elliptical contacts are described next

EXAMPLE

Find the contact parameters for two steel wires of the same diameter crossed at 90° Thisconfigurationisoftenusedinfretting wear studies Thenormal force is W = 5 [N], radii

of the wires are RA = RB = 1.5 × 10-3 [m], the Young's modulus for both wires is E = 2.1 ×

1011 [Pa] and the Poisson's ratio is υ = 0.3

· Reduced Radius of Curvature

Since the radii of the wires are Rax = ∞, Ray= RA = 1.5 × 10-3 [m], and Rbx = RB = 1.5 × 10-3 [m],

Rby= ∞ respectively, the reduced radii of curvature in the ‘x’ and ‘y’ directions are:

· Elliptical Contact Between Two Elastic Bodies, General Case

Elliptical contacts are found between solid bodies which have different principal relativeradii of curvature in orthogonal planes Examples of this are encountered in sphericalbearings and gears The contact area is described by an ellipse An illustration of this form ofcontact is shown in Figure 7.5 and the formulae for the main contact parameters aresummarized in Table 7.3

TABLE 7.3 Formulae for contact parameters between two elastic bodies; elliptical contacts,

Maximum contact pressure

Maximum deflection

W 2 E' 2 R'

( )1 /3

≈ 0.3p max

where:

a is the semimajor axis of the contact ellipse [m];

b is the semiminor axis of the contact ellipse [m];

R ' is the reduced radius of curvature [m];

k 1 , k 2 , k 3 , k 4 , k 5 are the contact coefficients

The other parameters are as defined previously Contact coefficients can be found from the

charts shown in Figures 7.12 and 7.13 [13] In Figure 7.12 the coefficients ‘k 1 ’, ‘k 2 ’ and ‘k 3’ are

plotted against the ‘k 0’ coefficient which is defined as:

φ is the angle between the plane containing the minimum principal radius of

curvature of body ‘A’ and the plane containing the minimum principal radius

of curvature of body ‘B’ For example, for a wheel on a rail contact φ = 90° while

for parallel cylinders in contact φ = 0°

The remaining contact coefficients ‘k 4 ’ and ‘k 5 ’ are plotted against the k 2 /k 1 ratio as shown inFigure 7.13

A very useful development in the evaluation of contact parameters is due to Hamrock andDowson [7] The method of linear regression by the least squares method has been applied to

derive simplified expressions for the elliptic integrals required for the stress and deflectioncalculations in Hertzian contacts The derived formulae apply to any contact and eliminatethe need to use numerical methods or charts such as those shown in Figures 7.12 and 7.13.The formulae are summarized in Table 7.4 Although they are only approximations, thedifferences between the calculated values and the exact predictions from the Hertziananalysis are very small This can easily be demonstrated by applying these formulae to thepreviously considered examples, with the exception of the two parallel cylinders In this casethe contact is described by an elongated rectangle and these formulae cannot be used Ingeneral, these equations can be used in most of the practical engineering applications.

1.5 2.0 0.5

1.0 1 2 5 10

(line contact) (point contact)

FIGURE 7.13 Chart for the determination of contact coefficients ‘k ’ and ‘k ’ [13]

TABLE 7.4 Approximate formulae for contact parameters between two elastic bodies [7].

Simplified elliptical integrals

Maximum contact pressure

Maximum deflection

a

b

4.5 εR'

[( )( 1 /3

b=( )6 πkE' εWR' 1 /3

W πkE')2

]Ellipticity parameter

ε and ξ are the simplified elliptic integrals;

k is the simplified ellipticity parameter The exact value of the ellipticity

parameter is defined as the ratio of the semiaxis of the contact ellipse in the

transverse direction to the semiaxis in the direction of motion, i.e k = a/b The differences between the ellipticity parameter ‘k’ calculated from the

approximate formula, Table 7.4, and the ellipticity parameter calculated from

the exact formula, k = a/b, are very small [7].

The other parameters are as defined already

EXAMPLE

Find the contact parameters for a steel ball in contact with a groove on the inside of asteel ring (as shown in Figure 7.7) The normal force is W = 50 [N], radius of the ball is

Rax = Ray = RA = 15 × 10-3 [m], the radius of the groove is Rbx = 30 × 10-3 [m] and the radius

of the ring is Rby = 60 × 10-3 [m] The Young's modulus for both ball and ring is E = 2.1 ×

1011 [Pa] and the Poisson's ratio is υ = 0.3

Since the radii of the ball and the grooved ring are Rax = 15 × 10-3 [m], Ray= 15 × 10-3 [m]and Rbx = -30 × 10-3 [m] (concave surface), Rby= -60 × 10-3 [m] (concave surface) respectively,

the reduced radii of curvature in the ‘x’ and ‘y’ directions are:

Since 1/R x < 1/R y condition (7.3) is not satisfied According to the convention it is

necessary to transpose the directions of the coordinates, so ‘R x ’ and ‘R y’ become:

R x = 0.02 [m] and R y = 0.03 [m]

and the reduced radius of curvature is:

· Maximum and Average Contact Pressures

= 1.6× 10 −6[m]

τmax= k4 p max = 0.33× 588.0 = 194.0 [MPa]

· Maximum and Average Contact Pressures

The benefits of applying the Hamrock-Dowson formulae to the evaluation of contactparameters are demonstrated by the simplification of the calculations without anycompromise in accuracy Hence the Hamrock-Dowson formulae can be used with confidence

in most practical engineering applications

Total Deflection

In some practical engineering applications, such as rolling bearings, the rolling element issqueezed between the inner and outer ring and the total deflection is the sum of thedeflections between the element and both rings, i.e.:

where:

δT is the total combined deflection between the rolling element and the inner and

outer rings [m];

δo is the deflection between the rolling element and the outer ring [m];

δi is the deflection between the rolling element and the inner ring [m]

According to the formula from Table 7.4, the maximum deflections for the inner and outerconjunctions can be written as:

where ‘i’ and ‘o’ are the indices referring to the inner and outer conjunction respectively.

Note that each of these conjunctions has a different contact geometry resulting in a different

reduced radius ‘R'’, ellipticity parameter ‘k’ and simplified integrals ‘ξ’ and ‘ε’

Introducing coefficients which are a function of the contact geometry and material properties,i.e.:

7.4 ELASTOHYDRODYNAMIC LUBRICATING FILMS

The term elastohydrodynamic lubricating film refers to the lubricating oil which separatesthe opposing surfaces of a concentrated contact The properties of this minute amount of oil,typically 1 [µm] thick and 400 [µm] across for a point contact, and which is subjected toextremes of pressure and shear, determine the efficiency of the lubrication mechanism underrolling contact

Effects Contributing to the Generation of Elastohydrodynamic Films

The three following effects play a major role in the formation of lubrication films inelastohydrodynamic lubrication:

· the hydrodynamic film formation,

· the modification of the film geometry by elastic deformation,

· the transformation of the lubricant's viscosity and rheology under pressure

All three effects act simultaneously and cause the generation of elastohydrodynamic films

· Hydrodynamic Film Formation

The geometry of interacting surfaces in Hertzian contacts contains converging and divergingwedges so that some form of hydrodynamic lubrication occurs The basic principles ofhydrodynamic lubrication outlined in Chapter 4 apply, but with some major differences.Unlike classical hydrodynamics, both the contact geometry and lubricant viscosity are afunction of hydrodynamic pressure It is therefore impossible to specify precisely a filmgeometry and viscosity before proceeding to solve the Reynolds equation Early attempts byMartin [2] were made, for example, to estimate the film thickness in elastohydrodynamiccontacts using a pre-determined film geometry, and erroneously thin film thicknesses werepredicted

· Modification of Film Geometry by Elastic Deformation

For all materials whatever their modulus of elasticity, the surfaces in a Hertzian contactdeform elastically The principal effect of elastic deformation on the lubricant film profile is

to interpose a central region of quasi-parallel surfaces between the inlet and outlet wedges.This geometric effect is shown in Figure 7.14 where two bodies, i.e a flat surface and a roller,

in elastic contact are illustrated The contact is shown in one plane and the contact radii are

‘∞’ and ‘R’ for the flat surface and roller respectively

h eA

Body A

Body B

FIGURE 7.14 Effects of local elastic deformation on the lubricant film profile

The film profile in the ‘x’ direction is given by [15]:

h = h f + h e + h g

where:

h f is constant [m];

h e is the combined elastic deformation of the solids [m], i.e h e = h e A + h e B;

h g is the separation due to the geometry of the undeformed solids [m], i.e for the

ball on a flat plate shown in Figure 7.14 h g = x 2 /2R;

R is the radius of the ball [m]

· Transformation of Lubricant Viscosity and Rheology Under Pressure

The non-conformal geometry of the contacting surfaces causes an intense concentration ofload over a very small area for almost all Hertzian contacts of practical use When a liquidseparates the two surfaces, extreme pressures many times higher than those encountered inhydrodynamic lubrication are inevitable Lubricant pressures from 1 to 4 [GPa] are found intypical machine elements such as gears As previously discussed in Chapter 2, the viscosity ofoil and many other lubricants increases dramatically with pressure This phenomenon isknown as piezoviscosity The viscosity-pressure relationship is usually described by amathematically convenient but approximate equation known as the Barus law:

ηp = η0e αp

where:

ηp is the lubricant viscosity at pressure ‘p’ and temperature ‘θ’ [Pas];

η0 is the viscosity at atmospheric pressure and temperature ‘θ’ [Pas];

α is the pressure-viscosity coefficient [m2/N]

As an example of the radical effect of pressure on viscosity, it has been reported that at contactpressures of about 1 [GPa], the viscosity of mineral oil may increase by a factor of 1 million(106) from its original value at atmospheric pressure [15]

With sufficiently hard surfaces in contact, the lubricant pressure may rise to even higherlevels and the question of whether there is a limit to the enhancement of viscosity becomespertinent The answer is that indeed there are constraints where the lubricant loses its liquidcharacter and becomes semi-solid This aspect of elastohydrodynamic lubrication is the focus

of present research and is discussed later in this chapter For now, however, it is assumedthat the Barus law is exactly applicable

Approximate Solution of Reynolds Equation With Simultaneous Elastic Deformation and Viscosity Rise

An approximate solution for elastohydrodynamic film thickness as a function of load, rollingspeed and other controlling variables was put forward by Grubin and was later superseded bymore exact equations Grubin's expression for film thickness is, however, relatively accurateand the same basic principles that were originally established have been applied in laterwork For these reasons, Grubin's equation is derived in this section to illustrate theprinciples of how the elastohydrodynamic film thickness is determined

The derivation of the film thickness equation for elastohydrodynamic contacts begins withthe 1-dimensional form of the Reynolds equation without squeeze effects (i.e 4.27):

dx = 6Uη h − h

h 3

( )where the symbols follow the conventions established in Chapter 4 and are:

p is the hydrodynamic pressure [Pa];

U is the surface velocity [m/s];

η is the lubricant viscosity [Pas];

h is the film thickness [m];

h is the film thickness where the pressure gradient is zero [m];

x is the distance in direction of rolling [m]

Substituting into the Reynolds equation the expression for viscosity according to the Baruslaw yields:

be approximated as a step jump to some value in pressure comparable to the peak Hertzian

contact pressure If this pressure is assumed to be large enough then the term e −αp « 1 and it can be seen from equation (7.17) that q ≈ 1/α Grubin reasoned that since the stresses and the

deformations in the EHL contacts were substantially identical to Hertzian, the opposingsurfaces must almost be parallel and thus the film thickness is approximately uniform

within the contact Inside the contact therefore, the film thickness h = constant so that h = h.

Since ‘h’ occurs where ‘p max’ takes place Grubin deduced that there must be sharp increase inpressure in the inlet zone to the contact as shown in Figure 7.15 It therefore follows that

according to this model q ≈ 1/α = constant, dq/dx = 0 and h = h within the contact.

Grubin’s model of contact pressure

p max p

Hertzian pressure

FIGURE 7.15 Grubin's approximation to film thickness within an EHL contact

A formal expression for ‘q’ is found by integrating (7.18);

h 1 is the inlet film thickness to the EHL contact [m];

h∞ is the film thickness at a distance ‘infinitely’ far from the contact [m]

Since q ≈ 1/α the above equation (7.19) can be written in the form:

remote from the contact, p = 0 and therefore q = 0 The following approximation was

calculated numerically for the integral as applied to a line contact:

R' is the reduced radius of curvature [m];

E' is the reduced Young's modulus [Pa];

L is the full length of the EHL contact, i.e L = 2l, [m];

b is the half width of the EHL contact [m];

h is the film thickness where the pressure gradient is zero, i.e Grubin's EHL film

thickness as shown in Figure 7.15 [m];

W is the contact load [N]

iterative numerical solution of the equations describing hydrodynamic film formation,elastic deformation and piezoviscosity in a lubricated Hertzian contact These are the samefundamental equations which are described above, but they are solved directly without anyanalytical simplifications The numerical procedures and mathematics involved aredescribed in detail in [7,11].

Pressure Distribution in Elastohydrodynamic Films

In a static contact, the pressure distribution is hemispherical or ellipsoidal in profileaccording to classical Hertzian theory The pressure field will change, however, when thesurfaces start moving relative to each other in the presence of a piezoviscous lubricant such

as oil Relative motion between the two surfaces causes a hydrodynamic lubricating film to

be generated which modifies the pressure distribution to a certain extent The greatestchanges to the pressure profile occur at the entry and exit regions of the contact Thecombined effect of rolling and a lubricating film results in a slightly enlarged contact area.Consequently at the entry region, the hydrodynamic pressure is lower than the value for adry Hertzian contact This has been demonstrated in numerous experiments The opposingsurfaces within the contact are almost parallel and planar and film thickness is often

described in this region by the central film thickness ‘h c’ The lubricant experiences aprecipitous rise in viscosity as it enters the contact followed by an equally sharp decline toambient viscosity levels at the exit of the contact To maintain continuity of flow andcompensate for the loss of lubricant viscosity at the contact exit, a constriction is formed close

to the exit The minimum film thickness ‘h 0’ is found at the constriction as shown in Figure7.16 The minimum film thickness is an important parameter since it controls the likelihood

of asperity interaction between the two surfaces Viscosity declines even more sharply at theexit than at the entry to the contact A large pressure peak is generated next to the constriction

on the upstream side, and downstream the pressure rapidly declines to less than dry Hertzianvalues The peak pressure is usually larger than the maximum Hertzian contact pressure anddiminishes as the severity of lubricant starvation increases and dry conditions areapproached [7] The size and the steepness of the pressure peak depends strongly on thelubricant's pressure-viscosity characteristics

h 0

Contacting surfaces

pressure distribution

U

Elastohydrodynamic pressure

distribution

FIGURE 7.16 Hydrodynamic pressure distribution in an elastohydrodynamic contact; h c is the

central film thickness, h 0 is the minimum film thickness

The end constriction to the EHL film is even more distinctive for a ‘point’ contact, e.g twosteel balls in contact In this case the contact is circular and the end constriction has to be

curved in order to fit into the contact boundary This effect is known as the ‘horse-shoe’constriction and is shown later in Figure 7.22 which illustrates a plan view of the EHL film(as opposed to the side view shown in Figure 7.16) The minimum film thickness in a pointcontact is found at both ends of the ‘horse-shoe’ and at these locations the film thickness is

only about 60% of its central value.

Elastohydrodynamic Film Thickness Formulae

The exact analysis of elastohydrodynamic lubrication by Hamrock and Dowson [7,16]provided the most important information about EHL The results of this analysis are theformulae for the calculation of the minimum film thickness in elastohydrodynamic contacts.The formulae derived by Hamrock and Dowson apply to any contact, such as point, linear orelliptical, and are now routinely used in EHL film thickness calculations They can be usedwith confidence for many material combinations including steel on steel even up tomaximum pressures of 3-4 [GPa] [11] The numerically derived formulae for the central andminimum film thicknesses, as shown in Figure 7.16, are in the following form [7]:

h c is the central film thickness [m];

h 0 is the minimum film thickness [m];

U is the entraining surface velocity [m/s], i.e U = (U A + U B )/2, where the subscripts

‘A’ and ‘B’ refer to the velocities of bodies ‘A’ and ‘B’ respectively;

η0 is the viscosity at atmospheric pressure of the lubricant [Pas];

E ' is the reduced Young's modulus (7.6) [Pa];

R ' is the reduced radius of curvature [m];

α is the pressure-viscosity coefficient [m2/N];

W is the contact load [N];

k is the ellipticity parameter defined as: k = a/b, where ‘a’ is the semiaxis of the

contact ellipse in the transverse direction [m] and ‘b’ is the semiaxis in the