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Elastohydrodynamic Lubrication  In highly loaded contacts o Roller bearing line contacts o Ball bearing point contacts o Gear teeth line contacts  Very high lubricant pressures p, order of Hertz pressures: ~ 100 ksi ≈ 700 Mpa  Elevated temperatures T in lubricant  Consequences o Lubricant viscosity η = η(p,T) increases o Surfaces deform, altering film thickness h o Altered film thickness affects pressures p

Stribeck Curve

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 Unknowns o Pressures p o Film thickness h o Temperatures T  Requires simultaneous solution of o Reynolds equation o Deformation of bodies, often elastic o Energy equation

Geometry of Rollers:  Roller on flat with same separation function:

1 1 1 = + R R1 R2 

€

Equivalent roller radius:

R=

R1R2 R1 + R2

U1 − U 2 ΔU S= = Slip-roll ratio: U€1 + U 2 2U , 0 ≤ S ≤ ∞ • •

S = 0: pure rolling (U1 = U2), no slip S = ∞: pure slip (U1 = - U2), no rolling

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Elastohydrodynamic Lubrication between Rollers •

Geometric film thickness:

h(θ) = ho + R(1 – cos θ) • • •

Convergent & divergent profiles Possible cavitation from divergent profile Possible elastic deformation of bodies, due to high pressures

•

Possible density & viscosity changes of lubricant, due to high pressures

•

Local geometry:

ElastoHydrodyamic Lubrication: Formulations Outline • Modify Reynold s Equation • Film Thickness Equation • Oil (Lubricant) Rheology • Reynolds Equation & Solutions • Effects

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Reynold s Equation • Assumptions 2D: ∂/∂z = 0 Incompressible: ∂ρ ∂ζ = 0, ζ = x, z, t Film thickness: h = h(x) Pressure: p = p(x) Velocities: W1 = W2 = 0 

h3 ∂p



∂   = 12(V − V ) 2 1 ∂x η ∂x ∂(U1 + U2) ∂h + 6h +6(U1 − U2) ∂x ∂x 2

R U2 h*

ho

x U1

x*

dh dx = dh U • V2 − V1 = dh = dt dx dt dx 2

• Right Hand Side: d(U1 +U2 ) dh dh = 12U2 dx + 6(U1 − U2) dx + 6h dx   d = 6 dx h(U1 + U2)

• Reynold s Equation: 



 d  d  h3 dp  =6 h(U1 + U2) dx η dx dx 3

Reynold s for Roller R U2 h*

ho

x U1

x*

• Integrate (h∗ = arbitrary constant): dp h − h∗ = 6η(U1 + U2) dx h3 • dp/dx = 0 at h = h∗, where x = x∗ • About x = 0, film thickness ``flattens . Approximate: h∗ ≈ ho. • Pressure boundary conditions: p = pa (or p = 0) at x = ±∞ 4

Film Thickness for Roller R U2 h*

ho

x U1

x*

• Local coordinates: x = Rθ, for h/R  1 • Film thickness: h = hg + hd • Geometric film thickness hg : x x2 hg (x) = ho + R(1 − cos ) ≈ ho + R 2R 5

• Deformation film thickness hd, from relative squash of bodies: 





hd(x) = v1(x) − v1(0) + v2(x) − v2(0)



• From Flamant solution (2D Bousinesq): 

¯ vi(x) = − P (1+ν) (κ + 1) ln r − 2πE

r=

2y 2



r2

|

y=0 P =1

x2 + y 2, κ = 3 − 4ν 2

2

1−ν1 1−ν2 1  • Let x = x−ξ, P = p(ξ)dξ, E = E + E , 1 2

∞ vi(x − ξ)dξ, then: vi(x) = −∞ p(ξ)¯  ∞

hd(x) = −

4 |ξ−x| dξ p (ξ ) ln πE −∞ |ξ|

Young s modulus: Ei, Poisson s ratio: νi

ME 383S

Bryant

March 10, 2005 5

FLAMANT SOLUTION (2D) • Point force P, normal to semi-infinite elastic space • 2D: plane strain or plain stress P x

(x, y) u y v

• Elastic deformations: (u, v) along (x, y) P  2xy  P  2y 2  u=− (κ −1)θ − 2  , v = − (κ + 1)log r − 2  4πµ  r  4πµ  r  • Stresses: 2P  y y 3  2P  y 3  2P  xy 2  σ xx = −  2 − 4  ’ σ yy = −  4  ’ σ xy = −  4  π r r € π r  π r 

€

µ: elastic shear modulus, ν: Poisson's ratio Elastic modulus: E = 2(1 + ν)µ € Dundars constant: κ = 3€– 4ν, plane strain, = (3 – ν)/(1 + ν), plane stress

€

2

r = x 2 + y , tan θ = x/y

€

Oil Rheology: Viscosity • Viscosity: η = η(p, T ) • Pressure dependence: η = ηpeα(p−pr ) • Temperature dependence: η = ηT eβ(T −Tr ) • η = η(p, T ) = ηoeα(p−pr )eβ(T −Tr )

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Final Reynold s Equation x2 + h (x) d 2R

dp = 6η(U1 + U2)  3 dx 2 x + h (x) ho − 2R d |ξ−x| 4 ∞ • hd(x) = − πE −∞ p (ξ ) ln |ξ| dξ

• η = ηoeα(p(x)−pr )eβ(T −Tr ) • Nonlinear integral-differential equation! • Unknowns: ho, p = p(x) • Boundary conditions: p |x=±∞= pa 7

Solution • Eigenvalue problem: special value of parameter (eigenvalue) for solution to exist Eigenvalue: ho Eigenfunction: p = p(x) • Usually solved numerically (via finite differences)

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Solution: EHD Pressures

•

•

Pressure profile p = p(x): o Approximately Hertzian o Pressure “spike” near outlet Film thickness h: o Roller flattens about contact zone o Nearly constant film thickness o Slight dimple forms near pressure spike Pressure measurements confirm spike!

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EHD Measurements

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Film thickness: • Sapphire roller against steel roller • Polarized light through sapphire, observe inference fringes for film thickness Temperature measurements: • Infra-red through sapphire roller roller

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K. Yagi, K. Kyogoku & T. Nakamura: “Relationship between temperature distribution in EHL film and dimple formation,” Proc. 2004 ASME/STLE Int’l Jt. Tribology Conf.

(a) S = -3.0

(um = -0.67 m/s)

(c) S = -8.0

(um = -0.25 m/s)

(e) S = -18.0

(um = -0.11 m/s)

(g) S = infinity (um = 0)

(b) S = 3.0

(um = 0.67 m/s)

(d) S = 8.0

(f) S = 18.0 (um = 0.11 m/s)

(um = 0.25 m/s)

Figure 3 Interference fringe patterns of film thickness for various slip ratios (pm = 0.97 GPa, ∆u = 2.0 m/s, Sapphire-steel contacts, oil : P100)

Film thickness

150

Disk

100

2 1.5 1

50

0.5

200

Film thickness

250

2 1.5

100

1

50

0.5 Ball Disk

0

0 -300 -200 -100

0 100 200 300

x, µm Tfmax = 357 K 1 50 0.8 100 50 200 0.6 200 0.4 300 150 200 0.2 100 50 50 0 -300 -200 -100 0 100 200 300

Z = z/h

Z = z/h

2.5

150

Ball

0 -300 -200 -100

300

0 100 200 300

x, µm T = 352 K fmax 1 50 200 0.8 50 300 0.6 0.4 100 50 300 50 0.2 200 0 -300 -200 -100 0 100 200 300

x, µm

(a) S = -8.0 (um = -0.25 m/s)

0

x, µm

3 2.5 Oil film

200

2

Film thickness

150

Disk

1

100 50

1.5

0.5

Ball

0 -300 -200 -100

Film Thickness h, µm

200

250

3 Film thickness h, µm Temperature rise, K

2.5

Oil film

Z = z/h

250

300

3 Oil film

Film thickness h, µm Temperature rise, K

Temperature rise, K

300

0 100 200 300

0

x, µm T = 344 K fmax 1 150 100 50 0.8 50 0.6 200 300 0.4 50 100 0.2 100 150 0 -300 -200 -100 0 100 200 300 x, µm

(b) S = infinity (um = 0)

(c) S = 8.0

(um = 0.25 m/s)

Figure 4 Temperature distribution and film profile in center line along sliding direction (pm = 0.97 GPa, ∆u = 2.0 m/s, Sapphire-steel contacts, oil : P100) Passing through the center of the contact area, the film thickness increases markedly. The maximum surface temperature rises of both the surfaces are about 100 K and the profiles cross at the center of the area. The temperature of the oil film, which is higher than that of both the surfaces, increases markedly at the dimple zone and reaches 357 K. The temperatures of both the surfaces also increase at the dimple zone. When the slip ratio is infinite, the film thickness is still about 0.1 m at both the sides of the contact area and the maximum film thickness reaches 2.0 m at the center of the

contact area, where the oil temperature rise increases dramatically. For comparison between S = -8.0 and 8.0, it is worth noting that the film thickness for the slip ratio of -8.0 is above 0.1 m thicker than that for the slip ratio of 8.0. This difference of the thicknesses is thicker than the accuracy for the film thickness. In addition, the maximum temperature rise in the oil film in case of the slip ratio of -8.0 is also higher than that of 8.0 as shown in Fig.4 and the difference for the temperatures is smaller than the temperature variation during the experiments. 4

Copyright © 2004 by ASME

0.05

(b) ∆u = 5.0 m/s

(c) ∆u = 9.0 m/s

Figure 5 Shape variation of film thickness for various sliding speed (pm = 0.97 GPa, S = infinity, Sapphire-steel contacts, oil : P100) 3

Oil film 0.03

300 Traction coefficient Ball

200

0.02

0.01

100 Disk

2.0 m/s 2.5 Sliding speed = 1.0 m/s Film thickness h, µm

0.04

400

Traction coefficient µ

(a) ∆u = 1.0 m/s

Maximum temperature rise, K

500

0 0

6.0 m/s

2

4

6

8

0 10

Sliding speed, m/s

2 1.5

Figure 7 Maximum temperature rise and traction coefficient for various sliding direction (pm = 0.97 GPa, S = infinity, Sapphire-steel contacts, oil : P100)

9.0 m/s

1 0.5

2 123 N -200

-100

0

100

200

300

Film thickness h, µm

0 -300

x, µm

Figure 6 Variation of film profile in center line along sliding direction for various sliding speed (pm = 0.97 GPa, S = infinity, Sapphire-steel contacts, oil : P100) Figures 5 to 7 show a series of results for the influence of the sliding speed at the slip ratio of infinity and load of 123 N. The effects of the sliding speed on the traction coefficient, the maximum temperature rise of the oil film and both the surfaces are illustrated in Fig.7. It can be seen that the maximum temperature rise of both the surfaces has a constant value of about 100 K whereas the maximum temperature rise of the oil film increases and thus reaches above 400 K. The traction coefficient demonstrates a gradual decrease and thus reaches a constant value of 0.02 asymptotically with increasing sliding speed. It should be noted that the dimple varies in shape from an ellipse where the minor axis is aligned with the sliding direction, a circle and an ellipse where major axis is aligned with the sliding direction as the sliding speed increases in Fig.5. The depth of the dimple increases with increasing sliding speed to 2.0 m/s and when the sliding speed is beyond 2.0 m/s, the depth decreases gradually as seen in Fig.6. The film thicknesses at both the sides of the contact area increase and thus reach a constant value of about 0.6 m. This tendency of the variation of the film thickness has been obtained in the experimental results of Cameron [11], Dyson et al [14] and Shorgin et al [19] as averaged values. The effects of the load on the maximum temperature rises, the traction coefficient and the film thickness are illustrated in Fig.8 and 9. As the load is increased, the size and depth of the dimple increases according to Fig.8. The thicknesses of both the sides of the contact area do not vary and have a constant

1.5

74 N

1

0.5

0 -300

Load = 25 N

-200

-100

0

100

200

300

x, µm

Figure 8 Variation of film profile in center line along sliding direction for various loads (∆u = 9.0 m/s, S = infinity, Sapphire-steel contacts, oil : P100) value of about 0.6 m. The temperature rises increase with load and the increase in temperature rise of the oil film is much higher than that of the both the surfaces as shown in Fig.9. The traction coefficient decreases with increasing load. CaF2-steel contacts Figure 10 shows the interference fringe pattern of the film thickness, the profiles of the temperature rises and the film thickness and temperature distribution across the oil film for the CaF2-steel contacts with simple sliding for a disk surface speed ud of 0.8 m/s against the stationary ball surface, the mean Hertzian pressure pm of 0.36 GPa using Santotrac100 oil. It can be seen easily that the dimple exits at the center of the contact. The temperature rise of the disk surface, which has lower thermal conductivity and is moving, is higher than that of the ball surface. The oil temperature rise increases at the dimple zone and reaches 211 K. As is seen in Fig.10, it is found that the heat is conducted 5

Copyright © 2004 by ASME