- Email: [email protected]
Chapter 7 Solidification theory
Chapter 7 Solidification theory 7.1
Introduction
The basis for hydrodynamic lubrication theory was formed by Reynolds [Reynolds 18861 in 1886. He derived the differential equation for the pressure build-up in a thin lubricating oil film, assuming the oil to be Newtonian. Using this equation Grubin [Grubin 19491 in 1949 solved the pressure build-up in the oil film between two rollers, assuming them to have the same form as a dry Hertzian contact. Later, Petrusevich [Petrusevich 19511, Dowson and Higginson [Dowson 19591, and others solved the elastohydrodynamic problem of lubrication of two rollers, assuming the lubricant to be Newtonian. This assumption is good for most oils at low shearing rates and at low pressures, but at those high pressures appearing in ball and roller bearings all thick mineral oils will be solidified [Jacobson 19701. When the oil is solidified, the pressure build-up is given by the shear strength and the compressibility of the oil, and by the elastic properties of the bearing surfaces. The shear strength of the oil limits the pressure derivatives, thus the high pressure peak, predicted by some authors, cannot exist. For example, the curves of Petrusevich give shear stresses in the oil of the order 150 - 1000 MPa.
7.2
Notation
radius of Hertzian contact area, [m] non-dimensional radius of the Hertzian contact area length and width of the calculation area, [m] b non-dimensional length and width of the calculation area constant in viscosity expression, [Pa-'] B non-dimensional constant in viscosity expression Bo specific heat of the oil, [J/kg "C] c e internal energy, [J/kg] moduli of elasticity, IPa] E l , E2,Eoil E power loss, [W] Eo non-dimensional power loss
a
a.
89
90
CHAPTER 7. SOLIDIFICATION THEORY
Fl, Fa frictional forces, [N] constant in viscosity expression oil film thickness, [m] non-dimensional oil film thickness minimum oil film thickness, [m] oil film thickness at the boundary between sliding and non-sliding solidified oil, [m] meshpoint coordinates (1 - ~ ) / ( T E I[Pa-’] ), (1 - 4 ) / ( T & ) , [Pa-’] h + ka, [Pa-’] non-dimensional deformability of the contact constant in viscosity expression coordinate in the direction of sliding, [m] pressure, [Pa] non-dimensional pressure p x , p z pressure in the I- and z-directions, [Pa] stiffness of the solidified oil, [Pa] (AplAv) ( A p / h ~ ) ~non-dimensional stiffness of the solidified oil ~ S E pressure at the boundary between sliding and non-sliding solidified oil, [Pa] P load capacity, [N] Po non-dimensional load capacity p, solidification pressure, [Pa] PO, non-dimensional solidification pressure solidification pressure at temperature T,, [Pa] p,, non-dimensional solidification pressure at non-dimensional PO,, temperature= 0 qz,q, oil flow per unit length in the z- and z-directions, [m2/s] qoz, qoz non-dimensional oil flows rise of solidification pressure per temperature unit, [Pa/”C] R rise of non-dimensional solidification pressure per non-dimensional & temperature unit Rlx, Rzx radii of curvature in the z-direction, [m] Rlz, R z z radii of curvature in the z-direction, [m] R,, R, equivalent radii of curvature of the contact, [m] S constant in viscosity expression t ,T temperature, [“C] temperature at the contact inlet (ambient temperature), [“C] T, AT temperature rise, [“C] 2’0 non-dimensional temperature AT0 non-dimensional temperature rise oil velocity in the 2-direction, [m/s] U non-dimensional velocity in the z-direction Uo U1, Uz surface velocities, [m/s] U, surface velocity, [m/s]
7.3. LUBRICATION OF HEAVILY LOADED SPHERICAL SURFACES surface deformations, [m] coefficient of heat transfer, [W/m2 "C] h, non-dimensional coefficient of heat transfer oil velocity in the z-direction, [m/s] W Wo non-dimensional oil velocity in the z-direction 2,y, z coordinates, [m] 20, zo non-dimensional coordinates Axe, Azo non-dimensional length differences 7 coefficient of thermal expansion of the oil, ["C-'1 70 non-dimensional coefficient of thermal expansion for the oil 'I approach of the elastic bodies, [m] ro non-dimensional approach of the elastic bodies 6 coefficient of compressibility for the oil in the liquid state, [Pa-'] 60 non-dimensional coefficient of compressibility for the oil in the liquid state 71 oil viscosity, [Pa s] qa oil viscosity at the contact inlet, [Pa s] 710 non-dimensional oil viscosity 8 the part of the unit width of the cavitation boundary which is filled with oil 01,712
V,
&/RZ
Poisson's ratio p oil density, [kg/m3] pa oil density at contact inlet, [kg/m3] po non-dimensional oil density T,,, T ~ , T , ~ , T,, , T,, ~ 1 , 7 2 shear stresses, [Pa] T~ shear strength of the solidified oil, [Pa] 7Oa non-dimensional shear strength of the solidified oil U
Indices a
m S
0
7.3 7.3.1
ambient mean value solidification non-dimensional
Lubrication of heavily loaded spherical surfaces General considerations
In 1886 Reynolds [Reynolds 18861 derived the partial differential equation
a
a
) + -(--) z(-7) a x az ph3ap
ph3ap 7) a z
= ~ ( Ut I U2)-
a(Ph) dX
91
CHAPTER 7. SOLIDIFICATION THEORY
92
Liquid oil
Figure 7.1 Lubricated circular contact.
for the pressure in a thin lubricating oil film, between two moving surfaces. The oil film thickness in equation 7.1 is assumed to be independent of time. Equation 7.1 is valid as long as the lubricant is Newtonian, but when the pressure reaches a certain value, the oil converts from a liquid to a solid. In this analysis this solidification is assumed to be instantaneous, i.e. the pressure increase to convert the oil from liquid to solid state is zero at the solidification pressure. The solidification pressure is highly dependent on the temperature. This wils shown in the report "Viscosity of Lubricants Under Pressure" by Hersey and Hopkins [Hersey 19541 where they gave the solidification pressure for the oil P62 at three different temperatures: 25 "C : 110 to 165 MPa 40 "C : 215 to 295 MPa 75 "C : 550 to 570 MPa These pressures are much lower than the maximum allowable pressure in a ball bearing. Modern ball bearings can withstand Hertzian pressures up to around 4 GPa if 90% of them have a lifetime of at least one million revolutions. If there are pressures higher than the solidification pressure at that temperature in a lubricated contact, the oil zone can be split up into four parts, see figure 7.1. These are liquid oil without cavitation, sliding solidified oil, non-sliding solidified oil, and the cavitation region.
7.3. LUBRICATION OF HEAVILY LOADED SPHERICAL SURFACES
t
93
y2
Figure 7.2 Coordinate system.
7.3.2
Elastohydrodynamic lubrication at relatively low pressures
This lubrication theory can only be used at relatively low pressures because of the solidification phenomenon. The gap height h is a function of the Cartesian coordinates x, z. In the general case the undeformed surfaces have the height function
where hmin = the minimum height (the minimum undeformed oil film thickness) and the R’s are the radii of curvature in the two coordinate directions. Because of the pressure in the oil film the two surfaces are deformed so that surface 1 gets the deformation q ( x , z ) in the yl-direction and surface 2 gets the deformation z ) ~ (zz), in the y2-direction, see figure 7.2. Simultaneously the centres of the spheres approach each other over the distance r. The total height (separation) at the point x, z is then
The deformation of the lubricated surfaces according to Hertz’ theory can be found in Timoshenko and Goodier “Theory of Elasticity” [Timoshenko 19551.
where A is the compressive load area.
To be able to solve the Reynolds equation one has to know how the density p and the viscosity 7 vary with pressure and temperature. Furthermore, the temperature field in the contact zone
CHAPTER 7. SOLIDIFICATION THEORY
94
T' "I
Figure 7.3 Oil element. has to be known. As the pressures in the liquid oil are comparatively low the simple density function P = P(P,T) = P d l - 7 A W + SP) is used, where 7 and 6 are positive constants, and pa is the lubricant density at the contact inlet. For the viscosity function the expression given in 1966 by C.J.A. Roelands [Roelands 19661 is chosen c[ 7 = qa10 (ltt1135)
( 4
where: qa =viscosity at atmospheric pressure and temperature 0°C p =pressure in MPa t =temperature in "C and G, M, S are constants Now study the distribution of temperature in the oil film. As the coefficient of thermal conductivity and the film coefficient of heat transfer between oil and metal are not known with any accuracy, the temperature is approximated to a constant through the oil film. The oil temperature minus the temperature of the bearing surfaces multiplied by a coefficient of heat transfer (Vm) gives the propagation of heat from the oil to the metal. The metal surfaces are rolling constantly so that new parts of the surfaces are coming into the contact region. Because of this the surface temperatures are supposed to remain constant. The power transferred to the oil element from the moving surfaces and from the surrounding oil elements, see figure 7.3, is given by
But the surface velocities are equal
u1 = u2 = ua
7.3. LUBRICATION OF HEAVILY LOADED SPHERICAL SURFACES and the oil flow per unit width in the two coordinate directions is
ap
h3 127 ax
qx = Uuh- --
h3 a p 127 az
qz =
This means that the mechanical power transferred to the oil element is
ap
a h3 a h3 ap dx dz dE” = ( 7 2 - 71)Uadx d z - { --[p(Uuh - -)I - --[p--1) ax 127ax az 12+ But the shear stress at the bottom surface is
and at the top surface
-
The velocity u of an oil element situated at a distance y from surface 1 is
a’(~’- yh) + (VZ- U l ) y / h+ UI
u = -I
27 a x
which gives the shear stress difference
and the power transmitted to the oil element
dE” = [-(--) a ph3ap a x 127ax
+ -(--) d ph3dp - V.p-1ah a2
12782
dxdz
dX
Heat transferred from the oil by conduction to the metal surfaces
dE’ = 2VmAT d x d z Continuity in heat for the element gives
a
a az
dE” = - ( q X p e ) dx d z + -(q,pe)
ax
dx d z + 2VmATdx d z
where: e is the internal energy of the oil given by e = cT and T is the temperature. The rules for partial derivatives give
95
CHAPTER 7. SOLIDIFICATION THEORY
96
solidified
01 I
I
oil s$difiec
ua
-X
Figure 7.4 Section of the oil film. where the continuity of mass gives
if the minimum oil film thickness is not changing with time. If the internal energy of the oil and the oil flows in the x- and z-directions are used in equation 7.2 the equation of heat transfer becomes
h3
apaT
1277
axax
-[Pc--
7.3.3
+
apaT
PC---]
azaz
- PCUah-
IT
ax
a ph3 ap + -(--) a ph3ap - Uap-ah - 2VmAT= 0 + A(--) a x 1277ax a z 12778% dX
Sliding solidified oil
The pressure build-up in the liquid region is determined by Reynolds’ equation, see figure 7.4. Study now the equilibrium of a solidified oil element with the cross section dxdz and the height h, see figure 7.5. In the x-direction:
If the terms containing r,, can be neglected and if the strength of the oil can be neglected in comparison to the oil pressure we get
7.3. LUBRICATION O F HEAVILY LOADED SPHERICAL SURFACES
I pdx dz
97
I p dxdz
Figure 7.5 Equilibrium of a sliding oil element.
These equations are the same as the equations for a liquid oil. The yield point for the solidified oil is assumed to be a constant for each oil, because the sliding solidified zone contains only a small pressure variation and therefore also a small shear strength variation. This constant . the oil elements are moving in relation to the surfaces, and the shear is denoted by T ~ As strength T~ is approximately constant, the above equation can be written in the form
The maximum pressure derivative is
and the oil elements are moving in relation to the metal surfaces so that the vectorial sum of the shear stresses equals the shear strength of the solidified oil.
The pressure in the solidified sliding zone is given by the above equation and the boundary conditions, which give continuity of mass.
CHAPTER 7. SOLIDIFICATION THEORY
98
Figure 7.6 Oil element.
7.3.4
Non-sliding solidified oil
The linear dimensions of the zone with non-sliding solidified oil are much larger than the oil film thickness. Therefore the deformations in the x- and z-directions can be neglected when the oil is compressed. The rise in pressure will then be proportional to the change in oil film thickness. p-PSE=--
~ S E h hSE
Eoir
1
- 2p 1-u
But if solidified oil is compressed in a cylinder and the unit volume is compressed by Av when the pressure rise is Ap, the modulus of elasticity for the oil Eo;r is
Ap 2ua Eo;l = -(1 - -) AV 1-u
Here, hSE is the oil film thickness at the inlet of the non-sliding solidified region. The pressure at the same point is p s ~This . equation has different values of p s and ~ h S E for different values of 2.
7.3.5
Solidification boundary conditions
For steady state conditions in the solidified zone the continuity of mass flow means that the amount of oil coming into a control volume must be equal to the amount leaving the same control volume per unit time. As the oil is solidified, the velocity components are constant through the thickness of the oil film, U and W respectively. The continuity of mass flow gives (7.5) This equation is valid in the whole solidified zone, see figure 7.6, but in the non-sliding part of it, W = 0 and U = U. = the velocity of the surfaces in the x-direction. The equation 7.5 then becomes
7.3. LUBRICATION OF HEAVILY LOADED SPHERICAL SURFACES
99
Figure 7.7 Sliding velocities and shear stresses.
ph = constant In the sliding zone the oil elements are moving in relation to the metal surfaces, in that direction which is given by the maximum pressure derivative at the point. Fkom uniform triangles in figure 7.7 :
u-u, - -71. -w
Tz
w = (U - Ua)-
7-2
71.
but from the equations 7.3 and 7.4
If this is inserted into equation 7.5, the result is:
-(pUh) a 8X
+ -a( p ( U at
- U.)-h)aPlaz
=0
aP/ax
This equation gives the velocity of the oil in the sliding zone if the boundary between the sliding and the non-sliding zones is known, because here U = U, and W=O. From the continuity of mass at the boundary between liquid oil and sliding oil it follows that (see figure 7.8):
CHAPTER 7. SOLIDIFICATION THEORY
100
X
Figure 7.8 Boundary between liquid and solidified oil. dz _ --qz - hW
dx q x - hU But the respective lubricant flows per unit width are in the x-direction
h3 a p qx = Uah - -128 a x and in the z-direction qz =
h3 a p
12q aZ
This gives the slope of the solidification boundary
If the expression 7.6 is substituted into the above equation, the continuity of mass at the boundary between liquid oil and solidified oil is given by
In each position along the solidification boundary the above equation has to be satisfied.
7.3.6
Cavitation theory
The boundary conditions for the liquid oil zone are determined by using the continuity of mass flow through the boundary between the oil zone and the cavitation zone, see figure 7.9 In the cavitation zone there are supposed to be an infinite number of thin oil streamers per unit width, which means that there are no sub-cavity pressures in the oil. Then the pressure
7.3. LUBRICATION OF HEAVILY LOADED SPHERICAL SURFACES
101
. X
Figure 7.9 Cavitation boundary.
in the cavitation zone is constant and equal to zero, that is, no pressure flow occurs. The stream of oil to the left of the cavitation boundary in figure 7.9 is then q1xA.Z
+ q1zAx
and to the right of the cavitation boundary
+
q2xAz 92rA.z The oil flows in the liquid non-cavitating region are per unit width in the x- and z-direction
h3 d p - --. 12q ax’
ap
= --12rlaz and in the cavitation region where the pressure is constant 91s = U,h
QlZ
h3
ow;
qZx = qZ2 = 0 where 0 is the part of the unit width at the cavitation boundary filled with oil
o c e g There is no oil produced or destroyed at the cavitation boundary, so the oil flow coming in must be equal to what flows out
_- _ _ _
U,h - h3 a p - h3 a p A x - OU,h 12q ax 12rlat
+0
h3 ap a p A x (1 - B)U,h - -(-1217 azaZ) = 0 Here, U,h > 0 and (1 - 0) 2 0 and because p = 0 at the start of the cavitation zone and p 2 0 in the whole contact the pressure gradient in the x-direction
ax +
C H A P T E R 7. SOLlDlFlCATlON THEORY
102
-aP- < o
ax -
and the pressure flow term in the z-direction
ap hX -50 az A~
i.e. the sum of three positive terms equals zero, so each term has to be eero.
The boundary condition for the cavitation region becomes
The boundary condition at the inlet is p=O
7.3.7
at x = - - o o .
Conversion to non-dimensional groups
To determine the pressure field and the height function between two lubricated surfaces rolling against each other, the equations needed are: Reynolds’ equation, the height function, the density function, the equation for heat transmission, and the viscosity function. Now introduce the following non-dimensional groups. Non-dimensional film thickness:
ho = -*
h
hmin
1
h0,min =
Non-dimensional x-coordinate: 20
Non-dimensional t-coordinate:
Non-dimensional pressure:
=
X
c
1
7.3. LUBRICATION OF HEAVILY LOADED SPHERICAL SURFACES Non-dimensional compressibility function:
Non-dimensional temperature:
T o
-
pncJh3
'
m'n T V n u n a
Non-dimensional coefficient of thermal expansion: 70 =
va
una
P
O
C
x K
7
The non-dimensional coefficient of heat transfer:
The function of heat transmission with the above non-dimensional groups is
With the same non-dimensional groups Reynolds' equation becomes
a
12-((1 -7oATo)(l +6opo)ho) axo In this notation the non-dimensional film thickness equation reads
where
and the approach of the two bodies
r ro= hmin
103
CHAPTER 7. SOLIDIFICATION THEORY
104
The dimensionless load carrying capacity of the contact is
The dimensionless oil flows in the x-and z-directions in the non-solidified zone qoz
h30 aP0 qz - ho - -= -Uahmin 1290 8x0
qoz =
qz hi PO = ---
Uahmin 12710~ In the solidified zone the non-dimensional oil flows are
qos
=q2 - Uoho
qoz
=qz - Woho -
Z O
Uahmin
Uahmin
The non-dimensional power loss in the lubricated contact is
The non-dimensional maximum shear stress in the sliding solidified zone is
The non-dimensional rise of the solidification pressure with temperature is
7.3.8
Numerical solution of the differential equations
The differential equations for the pressure and temperature are too complicated to be solved analytically. They are solved numerically by the method of finite differences. Reynolds' equation in dimensionless form is
a
poh:apo a poh:apo = 12-(poho) a t -(--) ax0 70 ax0 a20 To a20 ax0
-(--)
(7-9)
where PO
= (1 - yoATo)(1
+ 600~0)
By approximating the function with a parabola through three neighbouring points in the calculation grid, the derivatives in equations 7.9 and 7.8 can be approximated with the following expressions, see figure 7.10.
7.3. LUBRICATION OF HEAVILY LOADED SPHERICAL SURFACES
105
Figure 7.10 Square-net.
In the same way the derivative in the z-direction can be derived:
If the above approximations are used, Reynolds' equation in dimensionless difference form becomes
CHAPTER 7. SOLIDIFICATION THEORY
106
The heat transfer equation can also be rewritten in the same way: Poh;
Bitlj
[ ~ l i , j (
- Poi-l,j Toi+l,j- T'i-l,j
AXO
AXO
+
poi,j+l
- Bi,j-1 Toi,j+l - T01,3-1 ..
AZO
Azo
1-
To make it possible to solve this problem numerically, the zone of liquid oil has to be finite. Instead of two spheres in contact, two spherical surfaces, the sides of which are four times as large as the radius of the Hertzian contact area, are chosen. At the boundary of these spherical surfaces the pressure is put equal to zero. The radius of the contact area is
after Timoshenko and Goodier "Theory of Elasticity". If the dimensionless groups are used in in the above expression, the radius becomes in dimensionless form a0
=
a
37r
The length and width of the spherical surfaces in dimensionless form is then
a, = 4ao = y487rPOko In figure 7.11 a flow diagram is shown for the numerical calculations of the film pressure and the deformation of the spherical surfaces, which gives the oil film thickness and form. The numerical calchlations are made for the isothermal case with Ax0 = Azo = bo/10 and for a mineral oil. The figures 7.12 to 7.19 show the height functions and pressure fields for Po = 100 and 1000 for ICo = 0.001 and 0.1. In table 7.1 the parameter values and results for the calculated cases are given. The calculation of the power loss Eo converges very slowly, so these values are uncertain. , and l?o as function of ICO The figures 7.20 to 7.25 show diagrams for SO,7oa, pas, ( A p l A v ) ~BO and Po.
7.3. LUBRICATION OF HEAVILY LOADED SPHERICAL SURFACES
Start
Set a pressure field p i ,
Calculate elastic deformations. Add these to the undeformed height function. The sum =hi,
Calculate the pressure field in the liquid oil. qi, Calculate with the elasticity and plasticity conditions the pressure in the solidified oil. qi.j
Control the continuity at the solidification boundary.
I Change constants.
h I
Print p i , and hi,
Figure 7.11 Flow diagram for the numerical calculations.
107
108
CHAPTER 7. SOLIDIFICATION THEORY
Figure 7.12 Theoretical height function for Po = 100, ICO = 0.001, 60 = 9.85 x lod4, T~~= 630, pos = 61, (Ap/Av)o = 1.62 x lo5, Bo = 8.37 x ro= 0.368, Eo = 0.13.
t"
Figure 7.13 Theoretical pressure field for Po = 100, ko = 0.001, b0 = 9.85 x lo-*, rOs= 630, poS = 61, (Ap/A,v)o = 1.62 x lo5, BO = 8.37 x ro = 0.368, EO = 0.13. The broken line surrounds the solidified region.
7.3. LUBRICATION OF HEAVILY LOADED SPHERICAL SURFACES
109
Figure 7.14 Theoretical height function for PO = 1000, ko = 0.001, 60 = 4.50 x = 134, (Ap/Au)o = 3.6 x lo5, Bo = 3.81 x ro= 1.70, Eo = 11.
7oS = 4770, PO.,
Figure 7.15 Theoretical pressure field for Po = 1000, ko = 0.001, b0 = 4.50 x lo-4, T~~= 4770, = 134, (Ap/Av)o = 3.6 x lo5, BO = 3.81 x I'O = 1 . 7 0 , & , = 11. The broken line
surrounds the solidified region.
110
CHAPTER 7. SOLlDlFlCATlON THEORY
Figure 7.16 Theoretical height function for PO = 100, ICo = 0.1, 60 = 2.93 x r,, = 8.08, E~ = 6.3.
pOs = 1.59, ( A P / A V )= ~ 5.4 x 103, B~ = 0.249.
T~~ =
28.6,
Figure 7.17 Theoretical pressure field for Po = 100, ICO = 0.1, 60 = 2.93 x TO^ = 28.6, poS = 1.59, (Ap/Av)o= 5.4 x lo3, BO = 0.249. ro = 8.08, EO= 6.3. The broken line surrounds the solidified region.
7.3. LUBRICATION OF HEAVILY LOADED SPHERICAL SURFACES
111
Figure 7.18 Theoretical height function for Po = 1000, ko = 0.1, So = 0.015, T~~ = 428, 104, B~= 0.127. ro= 37.4,E~ = 50.
PO8 = 4.00, ( A ~ / A V ) ~ = 1.08 x
Figure 7.19 Theoretical pressure field for Po = 1000, ko = 0.1, So = 0.015, ros = 428, pos = 4.00, (Ap/Av)o = 1.08 x lo4, BO = 0.127. I’o = 37.4, Eo = 50. The broken line surrounds the solidified region.
CHAPTER 7. SOLIDIFICATION THEORY
112
0,001
0.1
0.01
Figure 7.20 Liquid lubricant compressibilitybO = qaUa
as function o f ko and PO.
1500 C
S
1000
500
0 0,001
0.01
Figure 7.21 Solidified lubricant shear strength ros = &rs
k0
0.1
as a function of ko and Po
7.3. LUBRICATION OF HEAVILY LOADED SPHERICAL SURFACES
113
30 PO,
20
10
0 0,001
0.01
Figure 7.22 Solidification pressure pod = -$&pa
1.5.10
ko as a function of
0,1
ko and Po.
5
(AP /dVl0
1Ot
5.101
0 0,001
0,Ol
kcl
Figure 7.23 Solidified lubricant stiffness (Ap/Av)o = &Ap/Av
Po.
0.1
as function of ko and
CHAPTER 7. SOLlDlFICATlON THEORY
114
0.25
B0 0,20 0,15 0.10 0.05
0 0,001
0,1
0.01
JcB as function of ko and PO.
Figure 7.24 Viscosity pressure relation constant Bo = qaUa
r,
35
30 25 20 15
10
5
0 0,001
Figure 7.25 Ball centre approach ro =
0.01
& as function of ko and PO.
0,1
7.3. LUBRICATION OF HEAVILY LOADED SPHERICAL SURFACES
7.3.9
Calculated data
Table 7.1 Parameter values and results.
7.3.10
Example
Determine the minimum oil film thickness between a ball and a plate. Data: R,=5 mm U,=O.l m/s q.=O.Ol Pa s 6=6 x lo-'' Pa-' ~ , = 1 5x lo6 Pa p,=lOs Pa (Ap/Av)=2.7 x 10" Pa B=5.1 x lo-' Pa-' k=5.52 x lo-'* Pa-'
P=l N
With the above data, the values of the dimensionless groups are
PO = 2.82 x 106\lh,i,
poS = 1.41 x 1 0 ° K Trial and error gives for hmin = 2 x lo-' m
115
CHAPTER 7. SOLIDIFICATION THEORY
116
PO = 2.82 x 1 0 6 d m= 400 ko =
2.76 x 10-17 = 0.069 4 x 10-16
Direct reading through the diagram in figure 7.22 also gives pod = 4. For all other minimum oil film thicknesses, hmin, pod gets a value which does not coincide with the value solved for by using the equations. Only one true solution for the oil film thickness exists and that is the oil film thickness which gives the same parameter values from the diagram readings as from the equations.
Bibliography [Dowson 19591
Dowson, D., and Higginson, G.R., “A Numerical Solution to the Elastohydrodynamic Problem”, J. Mech. Eng. Sc. 1, pp. 1-6, 1959.
[Grubin 19491
Grubin, A. N., and Vinogradova I. E., Central Scientific Research Institute for Technology and Mechanical Engineering. Book 30. Moscow (D.S.I.R. Translation No. 337), 1949.
[Hersey 19541
Hersey, M.D., and Hopkins, V., “Viscosity of Lubricants under Pressure”, ASME Research. Comm. on Lubrication, New York (The ASME), 1954.
[Jacobson 19701
Jacobson B.O., “On the Lubrication of Heavily Loaded Spherical Surfaces Considering Surface Deformation and Solidification of the Lubricant”, Acta Polytechnica Scandinavica, Mech. Eng. Series No. 54.
[Petrusevich 19511 Petrusevich, A.I., “Fundamental Conclusions from the ContactHydrodynamic Theory of Lubrication”, IZO. Akad. Nauk SSSR (OTN) 2 p. 209, 1951. [Reynolds 18861
Reynolds, O., “On the Theory of Lubrication and its Application to Mr. Beauchamp Tower’s Experiments, Including an Experimental Determination of the Viscosity of Olive Oil”, Phil. Trans. Roy. SOC.177, p. 157, 1886.
[Roelands 19661
Roelands, C.J.A., “Correlational Aspects of the Viscosity-TemperaturePressure Relationship of Lubricating Oils”, Delft, 1966.
[Timoshenko 19551 Timoshenko, S., and Goodier, J.N., “Theory of Elasticity”, New York McGraw-Hill Book Company Inc., 1955.
117
Introduction
The basis for hydrodynamic lubrication theory was formed by Reynolds [Reynolds 18861 in 1886. He derived the differential equation for the pressure build-up in a thin lubricating oil film, assuming the oil to be Newtonian. Using this equation Grubin [Grubin 19491 in 1949 solved the pressure build-up in the oil film between two rollers, assuming them to have the same form as a dry Hertzian contact. Later, Petrusevich [Petrusevich 19511, Dowson and Higginson [Dowson 19591, and others solved the elastohydrodynamic problem of lubrication of two rollers, assuming the lubricant to be Newtonian. This assumption is good for most oils at low shearing rates and at low pressures, but at those high pressures appearing in ball and roller bearings all thick mineral oils will be solidified [Jacobson 19701. When the oil is solidified, the pressure build-up is given by the shear strength and the compressibility of the oil, and by the elastic properties of the bearing surfaces. The shear strength of the oil limits the pressure derivatives, thus the high pressure peak, predicted by some authors, cannot exist. For example, the curves of Petrusevich give shear stresses in the oil of the order 150 - 1000 MPa.
7.2
Notation
radius of Hertzian contact area, [m] non-dimensional radius of the Hertzian contact area length and width of the calculation area, [m] b non-dimensional length and width of the calculation area constant in viscosity expression, [Pa-'] B non-dimensional constant in viscosity expression Bo specific heat of the oil, [J/kg "C] c e internal energy, [J/kg] moduli of elasticity, IPa] E l , E2,Eoil E power loss, [W] Eo non-dimensional power loss
a
a.
89
90
CHAPTER 7. SOLIDIFICATION THEORY
Fl, Fa frictional forces, [N] constant in viscosity expression oil film thickness, [m] non-dimensional oil film thickness minimum oil film thickness, [m] oil film thickness at the boundary between sliding and non-sliding solidified oil, [m] meshpoint coordinates (1 - ~ ) / ( T E I[Pa-’] ), (1 - 4 ) / ( T & ) , [Pa-’] h + ka, [Pa-’] non-dimensional deformability of the contact constant in viscosity expression coordinate in the direction of sliding, [m] pressure, [Pa] non-dimensional pressure p x , p z pressure in the I- and z-directions, [Pa] stiffness of the solidified oil, [Pa] (AplAv) ( A p / h ~ ) ~non-dimensional stiffness of the solidified oil ~ S E pressure at the boundary between sliding and non-sliding solidified oil, [Pa] P load capacity, [N] Po non-dimensional load capacity p, solidification pressure, [Pa] PO, non-dimensional solidification pressure solidification pressure at temperature T,, [Pa] p,, non-dimensional solidification pressure at non-dimensional PO,, temperature= 0 qz,q, oil flow per unit length in the z- and z-directions, [m2/s] qoz, qoz non-dimensional oil flows rise of solidification pressure per temperature unit, [Pa/”C] R rise of non-dimensional solidification pressure per non-dimensional & temperature unit Rlx, Rzx radii of curvature in the z-direction, [m] Rlz, R z z radii of curvature in the z-direction, [m] R,, R, equivalent radii of curvature of the contact, [m] S constant in viscosity expression t ,T temperature, [“C] temperature at the contact inlet (ambient temperature), [“C] T, AT temperature rise, [“C] 2’0 non-dimensional temperature AT0 non-dimensional temperature rise oil velocity in the 2-direction, [m/s] U non-dimensional velocity in the z-direction Uo U1, Uz surface velocities, [m/s] U, surface velocity, [m/s]
7.3. LUBRICATION OF HEAVILY LOADED SPHERICAL SURFACES surface deformations, [m] coefficient of heat transfer, [W/m2 "C] h, non-dimensional coefficient of heat transfer oil velocity in the z-direction, [m/s] W Wo non-dimensional oil velocity in the z-direction 2,y, z coordinates, [m] 20, zo non-dimensional coordinates Axe, Azo non-dimensional length differences 7 coefficient of thermal expansion of the oil, ["C-'1 70 non-dimensional coefficient of thermal expansion for the oil 'I approach of the elastic bodies, [m] ro non-dimensional approach of the elastic bodies 6 coefficient of compressibility for the oil in the liquid state, [Pa-'] 60 non-dimensional coefficient of compressibility for the oil in the liquid state 71 oil viscosity, [Pa s] qa oil viscosity at the contact inlet, [Pa s] 710 non-dimensional oil viscosity 8 the part of the unit width of the cavitation boundary which is filled with oil 01,712
V,
&/RZ
Poisson's ratio p oil density, [kg/m3] pa oil density at contact inlet, [kg/m3] po non-dimensional oil density T,,, T ~ , T , ~ , T,, , T,, ~ 1 , 7 2 shear stresses, [Pa] T~ shear strength of the solidified oil, [Pa] 7Oa non-dimensional shear strength of the solidified oil U
Indices a
m S
0
7.3 7.3.1
ambient mean value solidification non-dimensional
Lubrication of heavily loaded spherical surfaces General considerations
In 1886 Reynolds [Reynolds 18861 derived the partial differential equation
a
a
) + -(--) z(-7) a x az ph3ap
ph3ap 7) a z
= ~ ( Ut I U2)-
a(Ph) dX
91
CHAPTER 7. SOLIDIFICATION THEORY
92
Liquid oil
Figure 7.1 Lubricated circular contact.
for the pressure in a thin lubricating oil film, between two moving surfaces. The oil film thickness in equation 7.1 is assumed to be independent of time. Equation 7.1 is valid as long as the lubricant is Newtonian, but when the pressure reaches a certain value, the oil converts from a liquid to a solid. In this analysis this solidification is assumed to be instantaneous, i.e. the pressure increase to convert the oil from liquid to solid state is zero at the solidification pressure. The solidification pressure is highly dependent on the temperature. This wils shown in the report "Viscosity of Lubricants Under Pressure" by Hersey and Hopkins [Hersey 19541 where they gave the solidification pressure for the oil P62 at three different temperatures: 25 "C : 110 to 165 MPa 40 "C : 215 to 295 MPa 75 "C : 550 to 570 MPa These pressures are much lower than the maximum allowable pressure in a ball bearing. Modern ball bearings can withstand Hertzian pressures up to around 4 GPa if 90% of them have a lifetime of at least one million revolutions. If there are pressures higher than the solidification pressure at that temperature in a lubricated contact, the oil zone can be split up into four parts, see figure 7.1. These are liquid oil without cavitation, sliding solidified oil, non-sliding solidified oil, and the cavitation region.
7.3. LUBRICATION OF HEAVILY LOADED SPHERICAL SURFACES
t
93
y2
Figure 7.2 Coordinate system.
7.3.2
Elastohydrodynamic lubrication at relatively low pressures
This lubrication theory can only be used at relatively low pressures because of the solidification phenomenon. The gap height h is a function of the Cartesian coordinates x, z. In the general case the undeformed surfaces have the height function
where hmin = the minimum height (the minimum undeformed oil film thickness) and the R’s are the radii of curvature in the two coordinate directions. Because of the pressure in the oil film the two surfaces are deformed so that surface 1 gets the deformation q ( x , z ) in the yl-direction and surface 2 gets the deformation z ) ~ (zz), in the y2-direction, see figure 7.2. Simultaneously the centres of the spheres approach each other over the distance r. The total height (separation) at the point x, z is then
The deformation of the lubricated surfaces according to Hertz’ theory can be found in Timoshenko and Goodier “Theory of Elasticity” [Timoshenko 19551.
where A is the compressive load area.
To be able to solve the Reynolds equation one has to know how the density p and the viscosity 7 vary with pressure and temperature. Furthermore, the temperature field in the contact zone
CHAPTER 7. SOLIDIFICATION THEORY
94
T' "I
Figure 7.3 Oil element. has to be known. As the pressures in the liquid oil are comparatively low the simple density function P = P(P,T) = P d l - 7 A W + SP) is used, where 7 and 6 are positive constants, and pa is the lubricant density at the contact inlet. For the viscosity function the expression given in 1966 by C.J.A. Roelands [Roelands 19661 is chosen c[ 7 = qa10 (ltt1135)
( 4
where: qa =viscosity at atmospheric pressure and temperature 0°C p =pressure in MPa t =temperature in "C and G, M, S are constants Now study the distribution of temperature in the oil film. As the coefficient of thermal conductivity and the film coefficient of heat transfer between oil and metal are not known with any accuracy, the temperature is approximated to a constant through the oil film. The oil temperature minus the temperature of the bearing surfaces multiplied by a coefficient of heat transfer (Vm) gives the propagation of heat from the oil to the metal. The metal surfaces are rolling constantly so that new parts of the surfaces are coming into the contact region. Because of this the surface temperatures are supposed to remain constant. The power transferred to the oil element from the moving surfaces and from the surrounding oil elements, see figure 7.3, is given by
But the surface velocities are equal
u1 = u2 = ua
7.3. LUBRICATION OF HEAVILY LOADED SPHERICAL SURFACES and the oil flow per unit width in the two coordinate directions is
ap
h3 127 ax
qx = Uuh- --
h3 a p 127 az
qz =
This means that the mechanical power transferred to the oil element is
ap
a h3 a h3 ap dx dz dE” = ( 7 2 - 71)Uadx d z - { --[p(Uuh - -)I - --[p--1) ax 127ax az 12+ But the shear stress at the bottom surface is
and at the top surface
-
The velocity u of an oil element situated at a distance y from surface 1 is
a’(~’- yh) + (VZ- U l ) y / h+ UI
u = -I
27 a x
which gives the shear stress difference
and the power transmitted to the oil element
dE” = [-(--) a ph3ap a x 127ax
+ -(--) d ph3dp - V.p-1ah a2
12782
dxdz
dX
Heat transferred from the oil by conduction to the metal surfaces
dE’ = 2VmAT d x d z Continuity in heat for the element gives
a
a az
dE” = - ( q X p e ) dx d z + -(q,pe)
ax
dx d z + 2VmATdx d z
where: e is the internal energy of the oil given by e = cT and T is the temperature. The rules for partial derivatives give
95
CHAPTER 7. SOLIDIFICATION THEORY
96
solidified
01 I
I
oil s$difiec
ua
-X
Figure 7.4 Section of the oil film. where the continuity of mass gives
if the minimum oil film thickness is not changing with time. If the internal energy of the oil and the oil flows in the x- and z-directions are used in equation 7.2 the equation of heat transfer becomes
h3
apaT
1277
axax
-[Pc--
7.3.3
+
apaT
PC---]
azaz
- PCUah-
IT
ax
a ph3 ap + -(--) a ph3ap - Uap-ah - 2VmAT= 0 + A(--) a x 1277ax a z 12778% dX
Sliding solidified oil
The pressure build-up in the liquid region is determined by Reynolds’ equation, see figure 7.4. Study now the equilibrium of a solidified oil element with the cross section dxdz and the height h, see figure 7.5. In the x-direction:
If the terms containing r,, can be neglected and if the strength of the oil can be neglected in comparison to the oil pressure we get
7.3. LUBRICATION O F HEAVILY LOADED SPHERICAL SURFACES
I pdx dz
97
I p dxdz
Figure 7.5 Equilibrium of a sliding oil element.
These equations are the same as the equations for a liquid oil. The yield point for the solidified oil is assumed to be a constant for each oil, because the sliding solidified zone contains only a small pressure variation and therefore also a small shear strength variation. This constant . the oil elements are moving in relation to the surfaces, and the shear is denoted by T ~ As strength T~ is approximately constant, the above equation can be written in the form
The maximum pressure derivative is
and the oil elements are moving in relation to the metal surfaces so that the vectorial sum of the shear stresses equals the shear strength of the solidified oil.
The pressure in the solidified sliding zone is given by the above equation and the boundary conditions, which give continuity of mass.
CHAPTER 7. SOLIDIFICATION THEORY
98
Figure 7.6 Oil element.
7.3.4
Non-sliding solidified oil
The linear dimensions of the zone with non-sliding solidified oil are much larger than the oil film thickness. Therefore the deformations in the x- and z-directions can be neglected when the oil is compressed. The rise in pressure will then be proportional to the change in oil film thickness. p-PSE=--
~ S E h hSE
Eoir
1
- 2p 1-u
But if solidified oil is compressed in a cylinder and the unit volume is compressed by Av when the pressure rise is Ap, the modulus of elasticity for the oil Eo;r is
Ap 2ua Eo;l = -(1 - -) AV 1-u
Here, hSE is the oil film thickness at the inlet of the non-sliding solidified region. The pressure at the same point is p s ~This . equation has different values of p s and ~ h S E for different values of 2.
7.3.5
Solidification boundary conditions
For steady state conditions in the solidified zone the continuity of mass flow means that the amount of oil coming into a control volume must be equal to the amount leaving the same control volume per unit time. As the oil is solidified, the velocity components are constant through the thickness of the oil film, U and W respectively. The continuity of mass flow gives (7.5) This equation is valid in the whole solidified zone, see figure 7.6, but in the non-sliding part of it, W = 0 and U = U. = the velocity of the surfaces in the x-direction. The equation 7.5 then becomes
7.3. LUBRICATION OF HEAVILY LOADED SPHERICAL SURFACES
99
Figure 7.7 Sliding velocities and shear stresses.
ph = constant In the sliding zone the oil elements are moving in relation to the metal surfaces, in that direction which is given by the maximum pressure derivative at the point. Fkom uniform triangles in figure 7.7 :
u-u, - -71. -w
Tz
w = (U - Ua)-
7-2
71.
but from the equations 7.3 and 7.4
If this is inserted into equation 7.5, the result is:
-(pUh) a 8X
+ -a( p ( U at
- U.)-h)aPlaz
=0
aP/ax
This equation gives the velocity of the oil in the sliding zone if the boundary between the sliding and the non-sliding zones is known, because here U = U, and W=O. From the continuity of mass at the boundary between liquid oil and sliding oil it follows that (see figure 7.8):
CHAPTER 7. SOLIDIFICATION THEORY
100
X
Figure 7.8 Boundary between liquid and solidified oil. dz _ --qz - hW
dx q x - hU But the respective lubricant flows per unit width are in the x-direction
h3 a p qx = Uah - -128 a x and in the z-direction qz =
h3 a p
12q aZ
This gives the slope of the solidification boundary
If the expression 7.6 is substituted into the above equation, the continuity of mass at the boundary between liquid oil and solidified oil is given by
In each position along the solidification boundary the above equation has to be satisfied.
7.3.6
Cavitation theory
The boundary conditions for the liquid oil zone are determined by using the continuity of mass flow through the boundary between the oil zone and the cavitation zone, see figure 7.9 In the cavitation zone there are supposed to be an infinite number of thin oil streamers per unit width, which means that there are no sub-cavity pressures in the oil. Then the pressure
7.3. LUBRICATION OF HEAVILY LOADED SPHERICAL SURFACES
101
. X
Figure 7.9 Cavitation boundary.
in the cavitation zone is constant and equal to zero, that is, no pressure flow occurs. The stream of oil to the left of the cavitation boundary in figure 7.9 is then q1xA.Z
+ q1zAx
and to the right of the cavitation boundary
+
q2xAz 92rA.z The oil flows in the liquid non-cavitating region are per unit width in the x- and z-direction
h3 d p - --. 12q ax’
ap
= --12rlaz and in the cavitation region where the pressure is constant 91s = U,h
QlZ
h3
ow;
qZx = qZ2 = 0 where 0 is the part of the unit width at the cavitation boundary filled with oil
o c e g There is no oil produced or destroyed at the cavitation boundary, so the oil flow coming in must be equal to what flows out
_- _ _ _
U,h - h3 a p - h3 a p A x - OU,h 12q ax 12rlat
+0
h3 ap a p A x (1 - B)U,h - -(-1217 azaZ) = 0 Here, U,h > 0 and (1 - 0) 2 0 and because p = 0 at the start of the cavitation zone and p 2 0 in the whole contact the pressure gradient in the x-direction
ax +
C H A P T E R 7. SOLlDlFlCATlON THEORY
102
-aP- < o
ax -
and the pressure flow term in the z-direction
ap hX -50 az A~
i.e. the sum of three positive terms equals zero, so each term has to be eero.
The boundary condition for the cavitation region becomes
The boundary condition at the inlet is p=O
7.3.7
at x = - - o o .
Conversion to non-dimensional groups
To determine the pressure field and the height function between two lubricated surfaces rolling against each other, the equations needed are: Reynolds’ equation, the height function, the density function, the equation for heat transmission, and the viscosity function. Now introduce the following non-dimensional groups. Non-dimensional film thickness:
ho = -*
h
hmin
1
h0,min =
Non-dimensional x-coordinate: 20
Non-dimensional t-coordinate:
Non-dimensional pressure:
=
X
c
1
7.3. LUBRICATION OF HEAVILY LOADED SPHERICAL SURFACES Non-dimensional compressibility function:
Non-dimensional temperature:
T o
-
pncJh3
'
m'n T V n u n a
Non-dimensional coefficient of thermal expansion: 70 =
va
una
P
O
C
x K
7
The non-dimensional coefficient of heat transfer:
The function of heat transmission with the above non-dimensional groups is
With the same non-dimensional groups Reynolds' equation becomes
a
12-((1 -7oATo)(l +6opo)ho) axo In this notation the non-dimensional film thickness equation reads
where
and the approach of the two bodies
r ro= hmin
103
CHAPTER 7. SOLIDIFICATION THEORY
104
The dimensionless load carrying capacity of the contact is
The dimensionless oil flows in the x-and z-directions in the non-solidified zone qoz
h30 aP0 qz - ho - -= -Uahmin 1290 8x0
qoz =
qz hi PO = ---
Uahmin 12710~ In the solidified zone the non-dimensional oil flows are
qos
=q2 - Uoho
qoz
=qz - Woho -
Z O
Uahmin
Uahmin
The non-dimensional power loss in the lubricated contact is
The non-dimensional maximum shear stress in the sliding solidified zone is
The non-dimensional rise of the solidification pressure with temperature is
7.3.8
Numerical solution of the differential equations
The differential equations for the pressure and temperature are too complicated to be solved analytically. They are solved numerically by the method of finite differences. Reynolds' equation in dimensionless form is
a
poh:apo a poh:apo = 12-(poho) a t -(--) ax0 70 ax0 a20 To a20 ax0
-(--)
(7-9)
where PO
= (1 - yoATo)(1
+ 600~0)
By approximating the function with a parabola through three neighbouring points in the calculation grid, the derivatives in equations 7.9 and 7.8 can be approximated with the following expressions, see figure 7.10.
7.3. LUBRICATION OF HEAVILY LOADED SPHERICAL SURFACES
105
Figure 7.10 Square-net.
In the same way the derivative in the z-direction can be derived:
If the above approximations are used, Reynolds' equation in dimensionless difference form becomes
CHAPTER 7. SOLIDIFICATION THEORY
106
The heat transfer equation can also be rewritten in the same way: Poh;
Bitlj
[ ~ l i , j (
- Poi-l,j Toi+l,j- T'i-l,j
AXO
AXO
+
poi,j+l
- Bi,j-1 Toi,j+l - T01,3-1 ..
AZO
Azo
1-
To make it possible to solve this problem numerically, the zone of liquid oil has to be finite. Instead of two spheres in contact, two spherical surfaces, the sides of which are four times as large as the radius of the Hertzian contact area, are chosen. At the boundary of these spherical surfaces the pressure is put equal to zero. The radius of the contact area is
after Timoshenko and Goodier "Theory of Elasticity". If the dimensionless groups are used in in the above expression, the radius becomes in dimensionless form a0
=
a
37r
The length and width of the spherical surfaces in dimensionless form is then
a, = 4ao = y487rPOko In figure 7.11 a flow diagram is shown for the numerical calculations of the film pressure and the deformation of the spherical surfaces, which gives the oil film thickness and form. The numerical calchlations are made for the isothermal case with Ax0 = Azo = bo/10 and for a mineral oil. The figures 7.12 to 7.19 show the height functions and pressure fields for Po = 100 and 1000 for ICo = 0.001 and 0.1. In table 7.1 the parameter values and results for the calculated cases are given. The calculation of the power loss Eo converges very slowly, so these values are uncertain. , and l?o as function of ICO The figures 7.20 to 7.25 show diagrams for SO,7oa, pas, ( A p l A v ) ~BO and Po.
7.3. LUBRICATION OF HEAVILY LOADED SPHERICAL SURFACES
Start
Set a pressure field p i ,
Calculate elastic deformations. Add these to the undeformed height function. The sum =hi,
Calculate the pressure field in the liquid oil. qi, Calculate with the elasticity and plasticity conditions the pressure in the solidified oil. qi.j
Control the continuity at the solidification boundary.
I Change constants.
h I
Print p i , and hi,
Figure 7.11 Flow diagram for the numerical calculations.
107
108
CHAPTER 7. SOLIDIFICATION THEORY
Figure 7.12 Theoretical height function for Po = 100, ICO = 0.001, 60 = 9.85 x lod4, T~~= 630, pos = 61, (Ap/Av)o = 1.62 x lo5, Bo = 8.37 x ro= 0.368, Eo = 0.13.
t"
Figure 7.13 Theoretical pressure field for Po = 100, ko = 0.001, b0 = 9.85 x lo-*, rOs= 630, poS = 61, (Ap/A,v)o = 1.62 x lo5, BO = 8.37 x ro = 0.368, EO = 0.13. The broken line surrounds the solidified region.
7.3. LUBRICATION OF HEAVILY LOADED SPHERICAL SURFACES
109
Figure 7.14 Theoretical height function for PO = 1000, ko = 0.001, 60 = 4.50 x = 134, (Ap/Au)o = 3.6 x lo5, Bo = 3.81 x ro= 1.70, Eo = 11.
7oS = 4770, PO.,
Figure 7.15 Theoretical pressure field for Po = 1000, ko = 0.001, b0 = 4.50 x lo-4, T~~= 4770, = 134, (Ap/Av)o = 3.6 x lo5, BO = 3.81 x I'O = 1 . 7 0 , & , = 11. The broken line
surrounds the solidified region.
110
CHAPTER 7. SOLlDlFlCATlON THEORY
Figure 7.16 Theoretical height function for PO = 100, ICo = 0.1, 60 = 2.93 x r,, = 8.08, E~ = 6.3.
pOs = 1.59, ( A P / A V )= ~ 5.4 x 103, B~ = 0.249.
T~~ =
28.6,
Figure 7.17 Theoretical pressure field for Po = 100, ICO = 0.1, 60 = 2.93 x TO^ = 28.6, poS = 1.59, (Ap/Av)o= 5.4 x lo3, BO = 0.249. ro = 8.08, EO= 6.3. The broken line surrounds the solidified region.
7.3. LUBRICATION OF HEAVILY LOADED SPHERICAL SURFACES
111
Figure 7.18 Theoretical height function for Po = 1000, ko = 0.1, So = 0.015, T~~ = 428, 104, B~= 0.127. ro= 37.4,E~ = 50.
PO8 = 4.00, ( A ~ / A V ) ~ = 1.08 x
Figure 7.19 Theoretical pressure field for Po = 1000, ko = 0.1, So = 0.015, ros = 428, pos = 4.00, (Ap/Av)o = 1.08 x lo4, BO = 0.127. I’o = 37.4, Eo = 50. The broken line surrounds the solidified region.
CHAPTER 7. SOLIDIFICATION THEORY
112
0,001
0.1
0.01
Figure 7.20 Liquid lubricant compressibilitybO = qaUa
as function o f ko and PO.
1500 C
S
1000
500
0 0,001
0.01
Figure 7.21 Solidified lubricant shear strength ros = &rs
k0
0.1
as a function of ko and Po
7.3. LUBRICATION OF HEAVILY LOADED SPHERICAL SURFACES
113
30 PO,
20
10
0 0,001
0.01
Figure 7.22 Solidification pressure pod = -$&pa
1.5.10
ko as a function of
0,1
ko and Po.
5
(AP /dVl0
1Ot
5.101
0 0,001
0,Ol
kcl
Figure 7.23 Solidified lubricant stiffness (Ap/Av)o = &Ap/Av
Po.
0.1
as function of ko and
CHAPTER 7. SOLlDlFICATlON THEORY
114
0.25
B0 0,20 0,15 0.10 0.05
0 0,001
0,1
0.01
JcB as function of ko and PO.
Figure 7.24 Viscosity pressure relation constant Bo = qaUa
r,
35
30 25 20 15
10
5
0 0,001
Figure 7.25 Ball centre approach ro =
0.01
& as function of ko and PO.
0,1
7.3. LUBRICATION OF HEAVILY LOADED SPHERICAL SURFACES
7.3.9
Calculated data
Table 7.1 Parameter values and results.
7.3.10
Example
Determine the minimum oil film thickness between a ball and a plate. Data: R,=5 mm U,=O.l m/s q.=O.Ol Pa s 6=6 x lo-'' Pa-' ~ , = 1 5x lo6 Pa p,=lOs Pa (Ap/Av)=2.7 x 10" Pa B=5.1 x lo-' Pa-' k=5.52 x lo-'* Pa-'
P=l N
With the above data, the values of the dimensionless groups are
PO = 2.82 x 106\lh,i,
poS = 1.41 x 1 0 ° K Trial and error gives for hmin = 2 x lo-' m
115
CHAPTER 7. SOLIDIFICATION THEORY
116
PO = 2.82 x 1 0 6 d m= 400 ko =
2.76 x 10-17 = 0.069 4 x 10-16
Direct reading through the diagram in figure 7.22 also gives pod = 4. For all other minimum oil film thicknesses, hmin, pod gets a value which does not coincide with the value solved for by using the equations. Only one true solution for the oil film thickness exists and that is the oil film thickness which gives the same parameter values from the diagram readings as from the equations.
Bibliography [Dowson 19591
Dowson, D., and Higginson, G.R., “A Numerical Solution to the Elastohydrodynamic Problem”, J. Mech. Eng. Sc. 1, pp. 1-6, 1959.
[Grubin 19491
Grubin, A. N., and Vinogradova I. E., Central Scientific Research Institute for Technology and Mechanical Engineering. Book 30. Moscow (D.S.I.R. Translation No. 337), 1949.
[Hersey 19541
Hersey, M.D., and Hopkins, V., “Viscosity of Lubricants under Pressure”, ASME Research. Comm. on Lubrication, New York (The ASME), 1954.
[Jacobson 19701
Jacobson B.O., “On the Lubrication of Heavily Loaded Spherical Surfaces Considering Surface Deformation and Solidification of the Lubricant”, Acta Polytechnica Scandinavica, Mech. Eng. Series No. 54.
[Petrusevich 19511 Petrusevich, A.I., “Fundamental Conclusions from the ContactHydrodynamic Theory of Lubrication”, IZO. Akad. Nauk SSSR (OTN) 2 p. 209, 1951. [Reynolds 18861
Reynolds, O., “On the Theory of Lubrication and its Application to Mr. Beauchamp Tower’s Experiments, Including an Experimental Determination of the Viscosity of Olive Oil”, Phil. Trans. Roy. SOC.177, p. 157, 1886.
[Roelands 19661
Roelands, C.J.A., “Correlational Aspects of the Viscosity-TemperaturePressure Relationship of Lubricating Oils”, Delft, 1966.
[Timoshenko 19551 Timoshenko, S., and Goodier, J.N., “Theory of Elasticity”, New York McGraw-Hill Book Company Inc., 1955.
117